MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America () = Europe () = Asia +5 () = 10 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 2. Using your chosen value for k, write your version of the three mathematical models. 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. Consumer Ratings (in %) x, size of files North America Europe Asia 100 400 Page 1 of 2 900 1,600 2,500 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. 5. Describe the transformation of the graph of Europe with North America. 6. Describe the transformation of the graph of Asia with North America. 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx Page 2 of 2 MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x )= k x Europe E ( x )= k +5 x Asia A ( x) = k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E ( x )= 500 +5 x Asia A ( x) = 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (From the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably worls the same with Word and Excel) then fill-up the cells of the tables: key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5) were keyed-in. The remaining ratings were obtained by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered, select cells B3, C3, and D3. The left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, forrnat the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then Copy and Paste the entire table from Calc to Writer. 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 50 50 40 45 Ratings (in %) 30 30 25 20 20 10 21.67 16.67 11.67 0 0 500 17.5 12.5 North America Europe Asia 15 10 7.5 5 1000 1500 2000 2500 3000 x, size of files The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and the \"Copy and Paste\" from the spreadsheet to this document. 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N ( x ) = E ( x )= x). 500 x gives 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = (N(x) + 5, x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x) = (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graph from LibreOffice Calc and GeogeBra are given below to help illustrate the answers. Consumer ratings (in %) 60 50 40 North America Europe Asia 30 Rating 20 10 0 0 20000 40000 60000 80000 -10 x, size in megabytes 100000 120000 Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, 500 = + x x 0+ N ( x ) = lim . x 0+ lim Similarly, 500 + 5 = + x x 0+ E ( x ) = lim x 0+ lim 500 5 = + x x 0+ A ( x ) = lim . x 0+ and lim That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large. The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, lim N ( x ) = lim x + x + 500 =0 . x That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, lim E ( x ) = lim x + x + 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, lim A ( x ) = lim x+ x + 500 5 = 0 + ( 5 ) = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x) = 500 5 => x 500 5 = 0 => x 500 = 5 => x 100 = 1 => x x = 100 => x = 1002 => x = 10,000 . 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx Sheet1 1 K=500 2 Consumer Ratings 60 55 3 50 Ratings (in %) Consumer ratings (in %) x, size of files North America Europe Asia 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1600 12.5 17.5 7.5 2500 10 15 5 50 45 40 30 30 21.667 25 17 20 16.667 20 10 11.667 7. 0 0 500 1000 1500 x, size of fi Consumer ratin Possible asym 60 50 40 Rating x, size of files 100 400 900 1600 2500 10000 25000 75000 100000 Consumer ratings (in %) North America Europe Asia 50 55 45 25 30 20 16.667 21.667 11.667 12.5 17.5 7.5 10 15 5 5 10 0 3.1622776602 8.16227766 -1.83772234 1.8257418584 6.825741858 -3.17425814 1.5811388301 6.58113883 -3.41886117 30 20 10 0 -10 0 20000 40000 60000 800 x, size in megabytes Page 1 Sheet1 Consumer Ratings (in %) vs. Size of files 55 50 45 000 30 North America Europe Asia 21.667 25 17.5 15 16.667 20 12.5 10 11.667 7.5 500 1000 1500 5 2000 2500 3000 x, size of files Consumer ratings (in %) Possible asymptotes North America Europe Asia 40000 60000 80000 100000 120000 x, size in megabytes Page 2 MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x)= k x Europe E( x )= Asia k +5 x A ( x )= k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E( x )= 500 +5 x Asia A ( x )= 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. Consumer ratings (in %) x, size of files North America Europe Asia 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1,600 12.5 17.5 7.5 2,500 10 15 5 This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (From the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably worls the same with Word and Excel) then fill-up the cells of the tables: key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5) were keyed-in. The remaining ratings were obtained by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered, select cells B3, C3, and D3. The left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, forrnat the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then Copy and Paste the entire table from Calc to Writer. 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 5 2000 2500 3000 x, size of files The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and the \"Copy and Paste\" from the spreadsheet to this document. 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N (x )= gives E( x )= (N(x) + 5, x). 500 x 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x )= (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graph from LibreOffice Calc and GeogeBra are given below to help illustrate the answers. Consumer ratings (in %) Possible asymptotes 60 50 Rating 40 North America Europe Asia 30 20 10 0 -10 0 20000 40000 60000 80000 x, size in megabytes 100000 120000 Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, lim N (x ) = lim x 0 + x 0 Similarly, lim E ( x) = lim x 0 + x0 + + 500 = + . x 500 + 5 = + x and lim A( x) = lim x 0 + x0 + 500 5 =+ . x That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large. The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, 500 =0 . x + x lim N (x) = lim x + That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, lim E( x ) = lim x + x + 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, 500 5 = 0 +(5) =5 . Hence, as x (the size of the downloads in x + x lim A ( x) = lim x + megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x )= 500 5 => x => x = 100 2 500 5 = 0 => x 500 = 5 => x 100 = 1 => x x = 100 => x = 10,000 . 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x)= k x Europe E( x )= Asia k +5 x A ( x )= k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E( x )= 500 +5 x Asia A ( x )= 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. Consumer ratings (in %) x, size of files North America Europe Asia 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1,600 12.5 17.5 7.5 2,500 10 15 5 This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (From the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably worls the same with Word and Excel) then fill-up the cells of the tables: key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5) were keyed-in. The remaining ratings were obtained by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered, select cells B3, C3, and D3. The left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, forrnat the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then Copy and Paste the entire table from Calc to Writer. 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 5 2000 2500 3000 x, size of files The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and the \"Copy and Paste\" from the spreadsheet to this document. 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N (x )= gives E( x )= (N(x) + 5, x). 500 x 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x )= (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graph from LibreOffice Calc and GeogeBra are given below to help illustrate the answers. Consumer ratings (in %) Possible asymptotes 60 50 Rating 40 North America Europe Asia 30 20 10 0 -10 0 20000 40000 60000 80000 x, size in megabytes 100000 120000 Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, lim N (x ) = lim x 0 + x 0 Similarly, lim E ( x) = lim x 0 + x0 + + 500 = + . x 500 + 5 = + x and lim A( x) = lim x 0 + x0 + 500 5 =+ . x That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large. The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, 500 =0 . x + x lim N (x) = lim x + That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, lim E( x ) = lim x + x + 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, 500 5 = 0 +(5) =5 . Hence, as x (the size of the downloads in x + x lim A ( x) = lim x + megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x )= 500 5 => x => x = 100 2 500 5 = 0 => x 500 = 5 => x 100 = 1 => x x = 100 => x = 10,000 . 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx \fPK########f I################geogebra_thumbnail.png##PK##############PK########f I################geogebra_javascript.jsK+K.SHOO,T##PK##7######## PK########f I################geogebra_defaults2d.xml[s8#_C2#H&vfg3;Lg_}0 ,_!#$#uttt$q)#XR### `l%jz4>}<@#0 8$jlY:`T#.H#2"#\\{3#2YgJE'|>vD#t@uR[HwUDNtu+= me^#R#tg}) #ZD0"##bd#llUE5#N#p# Q=r#;I}#2hV7+#gb_t=*Nj#&E0#X#s{#]lhFtS#btGXD T`RI( #/"? *( #ob? "`n#n# <_ m##kpp;#ak-<##? ozbmc#d hs?#}%[| 8}ue,(e#,5]#mt?###p e`d0g$#6^++ #jq##>%IPL"\\,& #^8 ##?vZi>vr&| 1+|/## )#]d)wly" #{)#,! u)Wtv^#vbf#bY[#\\#:?7[F#obev##ru#'5k#Wl#&zT##Y ? O2J#ir;o##C#[nU #} 6S\\w#Bm:mz###}>Igv{7z*aN4C#b#|#_bPB u4D;c4Hbq8'h#,#I(#t,4gYG#6o&[ru/*XKc# ?h]##^uO"C%#v#LV*#2k}fr{]4H$ %##FD ##,f#,#7rgN#B# Y2hpN2VxZeznV- QT`#ZyB#y(,k^#~a"#:###L^$]H2#%##.c####>###%DN#!I#A#DC"##1x^2#A0#####RJ1# #Q#|B#C|^##hcJ03I_ Rn)Dc#D#2##N# skDqH@n#>SWzCpt#=1>Y'N]#`h# n#W#:kYh\\3{#####t io#Hon!-m1nJ?@LN)| M@K#EQ"9"wVY)'<#^(#BQgG"g##$Q0#hN]#fmWE?55Rg<}Qw -s$b &? y3)*#H#d%#?+#}#2MDR#:;jZLL8n+#$ ?}0R`#a#OxE~J=7p,E#Ea2]Qg# <#$#xx*#;jsA;[8?b ?6k3 -#uY-Je.&4[wO#a<[| [u#Bbg #Q1K,ALwXX ;hN5o(d#[#E#h 6no(;##r2:j+Uu;#{#ra]u8##3?#f# ## 2)TWxG8##[Ez]:}FA *+RfC= }#Dzn(|aWh /#xw=F_v###n>CK6ilmSs`j#?(~1|R#+ :d #FDP#a:# %_##z#='yIE* V#2 lmuk?Qc#),##rA##+ ## ` v#7# #RC## [sq^#~_{*#~P##J:R=#!+e2 #UCtl$;##ro#]@>$8wYbxTM!|BUs!#x_#M0? #6#g#~u#[iq#4_>##!##kxY#V UV#POk##M/# nYk{#)G##_L#leB#^1|(\\3++z:E? `#G_=#_#!<#gS##Wp##-o#PK##|b"###>###PK##########f I##############################geogebra_thumbnail.pngPK##########f I7######################F###geogebra_javascript.jsPK##########f I'F###$###################geogebra_defaults2d.xmlPK##########f IF### ###################geogebra_defaults3d.xmlPK##########f I|b"###>### ################geogebra.xmlPK##########L######## \fSheet2 Consumer ratings (in %) Possible asymptotes 60 50 40 North America 30 Rating Europe Asia 20 10 y=5 y = -5 0 0 20000 40000 60000 80000 100000 -10 x, size in megabytes Page 1 120000 140000 160000 180000 200000 \fMATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x )= k x Europe E ( x )= k +5 x Asia A ( x) = k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E ( x )= 500 +5 x Asia A ( x) = 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. (This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (from the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably works the same with Word and Excel) then fill-up the cells of the tables in the following manner: Key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5). Obtain the remaining rating by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered in a cell, select cells B3, C3, and D3. Then left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, format the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then \"Copy and Paste\" the entire table from Calc to Writer.) 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 50 50 40 45 Ratings (in %) 30 30 25 20 20 10 21.67 16.67 11.67 0 0 500 17.5 12.5 North America Europe Asia 15 10 7.5 5 1000 1500 2000 2500 3000 x, size of files (The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and then \"Copy and Paste\" from the spreadsheet to this document.) 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N ( x ) = E ( x )= x). 500 x gives 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = (N(x) + 5, x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x) = (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graphs from LibreOffice Calc (graphs 1 and 3) and GeoGebra (2nd graph) are given below to help illustrate the answers. Consumer Ratings (in %) vs. Size of files 60 55 50 50 40 45 Ratings (in %) 30 30 25 20 20 10 21.67 16.67 11.67 0 0 500 17.5 12.5 7.5 North America Europe Asia 15 10 5 1000 1500 2000 2500 3000 x, size of files Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, 500 = + x x 0+ N ( x ) = lim . + x 0 lim Similarly, 500 + 5 = + x x 0+ E ( x ) = lim x 0+ lim 500 5 = + x x 0+ A ( x ) = lim . x 0+ and lim That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large.ical definition The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, lim N ( x ) = lim x + x + 500 =0 . x That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, lim E ( x ) = lim x + x + 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, lim A ( x ) = lim x+ x + 500 5 = 0 + ( 5 ) = 5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x) = 500 5 => x 500 5 = 0 => x => x = 1002 => x = 10,000 . 500 = 5 => x 100 = 1 => x x = 100 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx Sheet1 1 K=500 2 Consumer Ratings (in %) vs. Size of files 60 55 3 50 Ratings (in %) Consumer ratings (in %) Europe Asia x, size of files North America 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1600 12.5 17.5 7.5 2500 10 15 5 50 45 40 30 North America Europe Asia 30 25 21.667 17.5 20 20 10 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 5 2000 2500 3000 x, size of files Consumer ratings (in %) North America Europe Asia 50 55 45 25 30 20 16.667 21.667 11.667 12.5 17.5 7.5 10 15 5 5 10 0 3.1622776602 8.16227766 -1.83772234 1.8257418584 6.825741858 -3.17425814 1.5811388301 6.58113883 -3.41886117 1.1180339887 6.118033989 -3.88196601 Consumer ratings (in %) Possible asymptotes 60 50 40 30 Rating x, size of files 100 400 900 1600 2500 10000 25000 75000 100000 200000 20 Page 1 Rating Sheet1 20 10 y=5 y = -5 0 0 20000 40000 60000 80000 100000 -10 x, size in megabytes Page 2 120000 Sheet1 h America pe %) North America Europe Asia Page 3 Europe Sheet1 Asia 120000 140000 160000 180000 200000 s Page 4 Sheet2 Consumer ratings (in %) Possible asymptotes 60 50 40 North America 30 Rating Europe Asia 20 10 y=5 y = -5 0 0 20000 40000 60000 80000 100000 -10 x, size in megabytes Page 5 120000 140000 160000 180000 200000 Sheet2 Page 6 Sheet2 Page 7 MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x)= k x Europe E( x )= Asia k +5 x A ( x )= k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E( x )= 500 +5 x Asia A ( x )= 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. Consumer ratings (in %) x, size of files North America Europe Asia 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1,600 12.5 17.5 7.5 2,500 10 15 5 (This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (from the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably works the same with Word and Excel) then fill-up the cells of the tables in the following manner: Key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5). Obtain the remaining rating by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered in a cell, select cells B3, C3, and D3. Then left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, format the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then \"Copy and Paste\" the entire table from Calc to Writer.) 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 5 2000 2500 3000 x, size of files (The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and then \"Copy and Paste\" from the spreadsheet to this document.) 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N (x )= gives E( x )= (N(x) + 5, x). 500 x 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x )= (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graphs from LibreOffice Calc (graphs 1 and 3) and GeoGebra (2nd graph) are given below to help illustrate the answers. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 x, size of files 5 2000 2500 3000 Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, lim N (x ) = lim x 0 + x 0 + 500 = + . x Similarly, lim E ( x) = lim x 0 + x0 + 500 + 5 = + x and lim A( x) = lim x 0 + x0 + 500 5 =+ . x That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large.ical definition The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, lim N (x) = lim x + x + 500 =0 . x That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x + x lim E( x ) = lim x + megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, lim A ( x) = lim x + x + 500 5 = 0 +(5) =5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x )= 500 5 => x => x = 100 2 500 5 = 0 => x 500 = 5 => x 100 = 1 => x x = 100 => x = 10,000 . 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x)= k x Europe E( x )= Asia k +5 x A ( x )= k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E( x )= 500 +5 x Asia A ( x )= 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. Consumer ratings (in %) x, size of files North America Europe Asia 100 50 55 45 400 25 30 20 900 16.667 21.667 11.667 1,600 12.5 17.5 7.5 2,500 10 15 5 (This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (from the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably works the same with Word and Excel) then fill-up the cells of the tables in the following manner: Key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5). Obtain the remaining rating by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered in a cell, select cells B3, C3, and D3. Then left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, format the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then \"Copy and Paste\" the entire table from Calc to Writer.) 4. Draw the graphs of the three mathematical models using Excel or another graphing utility. Do this on the same coordinate system so that the functions can be easily compared. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 5 2000 2500 3000 x, size of files (The graph was done in LibreOffice Calc. You can create it by inserting an OLE object Chart (From the Main Toolbar: Insert Object OLE Object Chart) but this method is tedious. It is probably better to create a Chart drawn using a spreadsheet and then \"Copy and Paste\" from the spreadsheet to this document.) 5. Describe the transformation of the graph of Europe with North America. Each point of the graph of Europe can be obtained by adding the quantity 5% to the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Adding 5 (%) to N (x )= gives E( x )= (N(x) + 5, x). 500 x 500 +5 . So each point of the graph of Europe is of the form (E(x),x) = x 6. Describe the transformation of the graph of Asia with North America. Each point of the graph of Asia can be obtained by subtracting the quantity 5% from the y-coordinate (2nd coordinate) of each point of the graph of North America. This can be deduced from the formulas given in Number 2. above. Subtracting 5 (%) from N (x )= 500 x gives A ( x )= (A(x),x) = (N(x) - 5, x). 500 5 . So each point of the graph of Asia is of the form x 7. Do the graphs have horizontal asymptotes? How about vertical asymptotes? Explain your answers. Graphs from LibreOffice Calc (graphs 1 and 3) and GeoGebra (2nd graph) are given below to help illustrate the answers. Consumer Ratings (in %) vs. Size of files 60 55 Ratings (in %) 50 50 45 40 30 North America Europe Asia 30 25 20 20 10 21.667 17.5 15 16.667 12.5 10 11.667 7.5 0 0 500 1000 1500 x, size of files 5 2000 2500 3000 Strictly speaking, there are only two asymptotes in the graphs. These asymptotes are both horizontal : a horizontal asymptote (x-axis or y = 0) for the graph of North America and a horizontal asymptote (y = 5) for the graph of Europe. This is so since we limit ratings only to values between 0% and 100%, inclusive. The vertical asymptotes becomes apparent if we let the ratings grow indefinitely large. This occurs when the size of the downloads become smaller and smaller but greater than zero. Hence, the vertical asymptote cannot appear if we stop when the rating is already 100%, i.e., when N(x), E(x), or A(x) has reached 100%, as this happens even before x becomes very close to zero. The horizontal asymptotes appear if we let the size of the downloads grow indefinitely large. Hence, none of the asymptotes is clearly seen in the graph in No. 4. However, the graphs presented here (from LibreOffice Calc and GeoGebra show the possibility of a vertical asymptote (x = 0) for all three graphs, and that y = 0 (NHL) and y = 5 (EHL) are horizontal asymptotes (but not y = -5, AHL). The last line (AHL) is below the x-axis and becomes asymptotic for negative values of the rating A(X). The vertical asymptote is the vertical line x = 0 (which is the y-axis) for each of the three graphs if there is no limit of 100% to the ratings. This is so since, lim N (x ) = lim x 0 + x 0 + 500 = + . x Similarly, lim E ( x) = lim x 0 + x0 + 500 + 5 = + x and lim A( x) = lim x 0 + x0 + 500 5 =+ . x That is to say, as x (the size of the downloads in megabytes), becomes smaller and smaller (goes to zero from the right, i.e., 0 +) then the ratings become (positively and) infinitely large.ical definition The horizontal asymptote is the horizontal line y = 0 (which is the x-axis) for North America, y = 5 for Europe and would have been y = -5 for Asia if ratings can be negative. This is so since, for North America, lim N (x) = lim x + x + 500 =0 . x That is to say, as x (the size of the downloads in megabytes), becomes bigger and bigger (increase without bound i.e., +) then the ratings N(x) become closer and closer to 0. Similarly, for Europe, 500 + 5 = 0 + 5 = 5 . Hence, as x (the size of the downloads in x + x lim E( x ) = lim x + megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings E(x) become closer and closer to 5. Lastly, for Asia, lim A ( x) = lim x + x + 500 5 = 0 +(5) =5 . Hence, as x (the size of the downloads in x megabytes), becomes bigger and bigger (increase without bound, i.e., +) then the ratings A(x) become closer and closer to -5. However, this is not seen in the graph as the range of the ratings A(x) is only from 100% to 0%. As can be seen in No. 8., the graph of y = A(x) stops at (10000,0) since A(10000) = 0 (%). 8. At what size, x (in megabytes), will the consumer ratings be equal to 0 in Asia? Show all of the work details. That would be x = 10,000 Mb. This is so since, A ( x )= 500 5 => x => x = 100 2 500 5 = 0 => x 500 = 5 => x 100 = 1 => x x = 100 => x = 10,000 . 9. Why do you think it is important to study the consumer ratings of an Internet service provider (ISP)? Explain your answer. Most likely to use the info so as not to lose the current subscribers of the service and to gain more subscribers by providing the appropriate information as each download is being requested, e.g., messages as to how slow or fast the download may take, or what time it is best to use the service, or how lengthy the queue may be at a certain time, as these may the cause of the low ratings (subscribers caught unaware of difficulties that may ensue when they download). Information regarding consumer ratings can be used to improve the quality of the service provided. It can help the company determine how to optimally appropriate their computer technology resources by focusing their attention on those download sizes that have very low ratings. In so doing, they can avoid that possibility of subscribers transferring to another ISP provider in the hope of being able to avail of better services. References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx MATH133: Unit 4 Individual Project 2B A company wants to study how consumers would rate Internet service providers (ISP's) based on the size of files that they download. The following equations model the relationship between two variables from the result of a survey conducted in North America, Europe, and Asia. In the equations, x is the size of files to be downloaded (in megabytes), N(x) is the average ratings in North America (in percent), E(x) is the average ratings in Europe (in percent), and A(x) is the average ratings in Asia (in percent). The consumer ratings can range from 0% (unsatisfactory) to 100% (excellent). For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places. North America N (x )= k x Europe E ( x )= k +5 x Asia A ( x) = k 5 x 1. Based on the first letter of your last name, choose a value for k that you will use in the equations above. It does not necessarily have to be a whole number. First letter of your last name Possible values for k A-F 500-599 G-L 600-699 M-R 700-799 S-Z 800-899 Choose k = 500 to facilitate calculations. 2. Using your chosen value for k, write your version of the three mathematical models. North America N (x )= 500 x Europe E ( x )= 500 +5 x Asia A ( x) = 500 5 x 3. Complete the table below by calculating the consumer ratings based on the size of the data. Work must be shown to receive full credit. (This table was obtained using LibreOffice Calc. If you are using LibreOffice, you can insert an OLE object (from the Main Toolbar: Insert Object OLE Object LibreOffice 5.0 Spreadsheet; probably works the same with Word and Excel) then fill-up the cells of the tables in the following manner: Key-in the given of the question and the formula for the rating in the first three blank cells (=500/SQRT(A3), =500/SQRT(A3)+5, and =500/SQRT(A3)-5). Obtain the remaining rating by auto-filling: after the first 3 formulas (in cells B3, C3, and D3) have been keyed-in and the Enter key pressed after each formula was entered in a cell, select cells B3, C3, and D3. Then left-click without releasing on the little black square that appears in cell D3 and drag downwards until D7. These will generate the contents of the remaining empty cells. After the table is generated, format the answer cells with blue background.) Or, simply generate the table in LibreOffice Calc with steps similar to the ones described above, then \"Copy and Paste\" the entire table from Calc to Writer.) North America x,