Part A Fitting Exponential Curves to Data (possible 16 points} Cooling Tea Activity: You can use your graphing calculator to t an exponential curve to data and nd the exponential function. The table at the right shows the number of degrees above room temperature for a cup of tea after x minutes of cooling. Graph the data. Find the besttting exponential function. Step 1 Press STAT ENTER on your calculator to enter the data in lists. Step 2 Set up STAT PLOT lor Plot 1 to draw the scatter plot. Press 2ND Y=. select 0n, point graph, L1 for XList. and L2 for YList. Cooling Tea Time I\"F Above Room min Temperature 0 135 5 100 1 0 M 1 5 55 2 0 d1 2 5 30 30 22 35 1? 40 12 45 9 50 7 55 5 SD 4 Step 3 Find the equation for the best-fitting exponential function. Press STAT CALC Exp Reg. The line of ExpReg y=a*bAx best fit is estimated by f(x) = 133.458(0.942) a=133.4584506 b= . 942405561 r=-. 9997925841 Step 4 Graph f(x). Press Y= Clear VARS 5, arrow over twice and press ENTER to display the ExpReg results. Press GRAPH to display both the function and scatter plot together. To Zoom results, Press ZOOM 9. For additional information on exponential curves and line of best fit, please visit https://www.youtube.com/watch?v=K3HjLrg14DQ and https://www.youtube.com/watch?v=IE- FdDSJCPw Use a graphing calculator to find the exponential function that best fits each set of data. Graph each function. Sketch your graph. X -3 -2 -1 0 1 2 (3 pts) 50 25 12.5 6.25 3.13 1.56 2 X 0 1 2 3 4 5 (3 pts) V 2 2.4 2.88 3.46 4.15 5 3 X 1 2 3 4 5 6 (3 pts) V 1.2 4.8 19.2 76.8 307.2 1228.8 4 X -10 -9 -8 -7 -6 -5 (3 pts) V 0.03 0.07 0.14 0.27 0.55 1.095. In the Cooling Tea Activity mm at the beginning of this section, the function appears to level off. Explain why this happens. (1 pt] 6. a. Find a llnearfunctlon that models the data in Question 3 from above. {2 pts] b. Which is a better t. the linear function or the exponential tunctlon? Explain. [1 pt] Part B The pH Concept (possible 14 points) If you have taken Chemistry. you may recall molarlty or molar concentration. Molarlty is the number of moles of solute dissolved in 1 liter of solution. Expressing hydrogen-ion concentration [Ht] in molarin is not practical, therefore another system called the pH scale is used. The pH scale ranges from U to 14. A solution with a pH less than T" is acldlc. A solution with a pH greater than T is basic. The pH of a solution is the negative logarithm of the hydrogenion concentration. In pure water or a neutral solution, the [H1 = 1 x104 M, and the pH is ?. Activity: Use the information in the table to nd the pH or hydrogenion concentration of each solution (or food item) listed. Tell whether the solution (or food item) is basic or acidic. Round to the nearest hundredth. Fill inthe chart completely. [14 p3) pH = 409 [H*'.l = log[1x10_?) = (log 1 + (1?) log to) = (o + {4)} = 7 Foodl Solution [H'] pH Acid or base Soda (avg) 3.26 Fresn egg Unknown 12.6 Cfeam 2.5 x to" Ketchup 1 3 x 10-4 Unknown 5.00 Cractcers 43 x 10-9 Part C - Linear and Exponential Models (possible 15 points) You can transform an exponential function into a linear function by taking the logarithm of each side. Since linear models are easy to recognize, you can then determine whether an exponential function is a good model for a set of values. y = ab" Write the general form of an exponential function. log y = log ab* Take the logarithm of each side. log y = log a + x log b Product property and power property. log y = (log b)x + log a Rewrite. If log b and log a are constants, then log y = (log b)x + log a is a linear equation in slope-intercept form. To confirm that log b is a constant, check that the graph of log y = (log b)x + log a is a line Activity: Determine whether an exponential function X 0 2 6 8 10 is a good model for the values in the table. 3.0 5.1 8.6 14.5 24.5 41.4 Step 1 Enter the values into STAT lists L1 and Step 2 To graph log y, set-up STAT PLOT L2. To enter the values for log y, place feature and press 1. Then enter L3 cursor in the heading of L3 and press next to YList. To Zoom results, Press LOG 2ND 2 ENTER ZOOM 9. L2 L3 0.47712 5.1 0.70757 8.6 0.9345 14.5 1.1614 24.5 1.3892 41.4 1.617 L1(1) = 0 Since the graph of log y = (log b)x + log a is linear, the slope log b is constant, and b also is constant. An exponential function therefore is a suitable model. Step 3 Press STAT arrow over right 0 ENTER to find the exponential function; y = 3(13) For each set of values, determine whether an exponential function is a good model. If so, find the function. If not, explain why. X 5 (2 pts) 6 22 54 102 145 No because log a + log b isn't linearI "I 0 y 402 19.3 1 9.9 2 3 5.1 2.5 YES 3- x o 1 2 3 4 1 mm\"- ! -9 -? -5 -3 -1 1 -3 -2 -1 0 1 2 5- II"! 2 a II\" as [2 p15} [2 p15} [2 pls} [2 p15} 6. The value of a car depreciates over time. The table shows the value of a car over a gig period. {5 pts] :1 o 1 2 3 4 y 13,500 15,910 13,682.60 11.76104 10,119.55 3. Determine which Kind of function best models the data. 0. Write an equation for the function that models the data. C. Use your equation to determine how much the car is worth after 7 years. Part D Exponential and Logarithmic Inequalities [possible 20 points} Legend tells of a Roman general who was so successful in his campaigns that the emperor oered the general his choice of reward. The general asked the emperor tor all the silver that he could carry out of the imperial treasury in one month {31 days]. The emperor agte_ed,_h_ul offered two conditions. Oh the rst day, the ge heral would receive 1 silver denarius. On the second day, he would tum in the previous day's coin and receive a coin valued at 2 deharii. Each day. the general was to come to the treasury Hg in the previous day's coin and the treasurer would mint a special ooin that would be twice as heavy and worth twice as much as the coin the general got a day earlier. One denarius coin weighed about 0.006 kg. The largest object the general could carry or roll without help could weigh no more than 300 kg. Activity 1: 1. a. What function V00 gives the value of the coin that the general received on day I? (2 pts] in. What is the domain of this function? [1 pt] 2. Write a function M{x) that gives the mass of each coin in kilograms as a function of day x. [2 PB} 3. a. Use M(x) from Question 2 to write an inequality that describes the mass of a coin that the general could carry or roll out of the treasury. (2 pie] 0. Make a table of values of this function. [3 p15} c. Use the table to solve the inequality. [2 pls} 4. What is the total value of the coins that the general would receive? {1 pt) Activity 2: Scientists are growing bacteria in a laboratory. They start with a known population of bacteria and measure how long it takes this population to double. 5. Sample A starts with 200,000 bacteria. The population doubles every hour. Write an exponential function that models the population in Sample A as a function of time in hours. [1 Fl!) 6. Sample 5 starts with 50,000 bacteria. The population doubles every half hour. Write an exponential function that models the population growth in Sample 8 as a function of time in hours. {2 pts} I a. Write an inequality that models the population in Sample 8 overtaking the population in Sample A. [2 pts} b. Use a graphing calculator to solve the inequality. {2 pts] Part E Rational Inequalities (possible 35 points] When you solve rational W you can't necessarily solve them exactly like rational equations. If you multiply both sides of a rational inequality by the same algebraic expression just as you have done with equations, you can introduce extraneous solutions or lose solutions! For additional information on rational inequalities and how to solve rational inequalities by graphing. Use your Internet browser and please visit: htfosfmkhanacademy.orgfmafhfalgebrahomez'alg; ratronalexoreorunoralgratl'onalmeouallfiesraonall'neoualr'tl'es and hn'IMIl-ww. gallium. COMIC!) 7U=waC 5 'l l'HQl'g. Activity 1: '1- Here is Neelyis solution ofthe rational inequality