Math 1324 Lab 1 - Chapter 1 and 2 1.) A manufacturer's cost to manufacture 50 microwaves is $14,040, while the cost to make 400 microwaves is $28,040. Let x = number of microwaves and y = cost . a.) Assuming the cost function is linear find the cost function using algebraic methods. Show work. b.) From the cost function, state and interpret the average rate of change in the cost to make a microwave. c.) What are this manufacturer's fixed costs? _____________________ d.) This manufacturer will sell these microwaves for $79.00 each. State the linear revenue function. e.) Using algebraic methods, find the breakeven point for the microwave manufacturer. Show work. f.) Using a complete sentence, explain what this breakeven point means in this problem. Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t1 2.) Let the supply and demand for bananas in cents per pound be given by : supply: p = 52 q demand: p =100 - 52 q a.) Graph these equations on the same axes. b.) Find the equilibrium quantity and price. Use algebraic methods. c.) On what interval of the graph ( referring to "q") does demand exceed supply? 3.) If the supply function for a commodity is p = q2 + 8q + 16 and the demand function is p = -3q2 + 6q + 36, find the equilibrium quantity and equilibrium price. Show algebraic work. With a complete sentence explain the meaning of your answers. Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t2 4.) A company produces a certain toy and finds that the cost of producing x toys is given by the function C(x) = 1.50x + 7500. The revenue from selling x toys is given by the function R(x) = 6.50x. a.) What is the marginal cost? ______________ b.) What are the fixed costs for this company? ___________________ c.) What is the marginal revenue? _______________ d.) What will it cost the company to make 5000 of the toys? __________________ e.) What revenue will result from the sale of 375 of the toys?______________ f.) Find the break-even point algebraically. Show algebraic work. g.) Find the profit function for the sale of this toy. P(x) = _________________________________ h.) What is the marginal profit? _______________ 5.) A charter flight charges a fare of $200 per person, plus $4 for each unsold seat on the plane. The plane holds 100 people. (For example, if there were 10 unsold tickets, each ticket sold would cost $200 + 10 * $4 = $240) a.) If x represents the number of UNSOLD seats on the plane, what expression would represent the number of SOLD seats on the plane? b.) Write an equation that represents the price (p) of a seat on the charter plane. Let x = number of UNSOLD seats on the plane. c.) Write an equation for the total revenue (R) received for the flight. d.) Find the number of unsold seats that will produce the maximum revenue. Show work. e.) Find the maximum revenue. Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t3 6.) In a report entitled Projected Supply, Demand, and Shortages of Registered Nurses:2000-2020, figures were presented for the number of registered nurses that were "demanded" in the years 2000 to 2020. The data for 2000 - 2012 are presented below. Let x= number of years past 2000. Y= the demand column. *Note: As you put the y values into a list on the calculator they will be converted to scientific notation because of their size. Year x-value Demand 2000 0 1,999,950 2001 1 2,030,971 2002 2 2,062,556 2003 3 2,095,514 2004 4 2,128,142 2005 5 2,161,831 2006 6 2,196,904 2007 7 2,232,516 2008 8 2,270,890 2009 9 2,307,236 2010 10 2,344,584 2011 11 2,379,719 2012 12 2,426,741 2013 13 2,472,072 2014 14 2,516,827 a.) Enter the x and y data in two lists in your calculator and observe a scatterplot. Does there appear to be a linear trend? _________________ Is it an increasing or decreasing trend? ____________________ b.) Using the calculator, obtain the equation for the linear model. Round coefficients to 3 decimal places. Write it here: _____________________________________________ Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t4 7.) In the same study, the number of registered nurses that were being supplied was also given. That data is given in the table below. Let x= number of years past 2000. Y= the demand column. Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 x-value Supply 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1,889,243 1,912,667 1,937,336 1,959,192 1,989,329 2,012,444 2,028,548 2,039,772 2,047,729 2,059,099 2,069,369 2,075,891 2,075,218 2,068,256 2,061,348 a.) Enter the x and y data in two lists in your calculator. Obtain the equation for a linear model. Round coefficients to 3 decimal places. Write it here: _________________________________________________ b.) Use the linear regression equations for demand (in problem #6) and supply (#7 part a) to predict how many registered nurses will be "demanded" and "supplied" in 2015 and 2020. Use an x-value of 15 (representing 2015) and x-value of 20 (for 2010) Year x-value Nbr of Nurses Demanded Nbr of Nurses Supplied 2015 2020 c.) What observation can you make from these numbers with regard to possible future market equilibrium? Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t5 d.) In the "y=" area on your calculator, type the linear equation from #1 which models the demand function for registered nurses. In the 2nd line, type the quadratic equation above which models the supply function. Change your window to these settings: x-min = 0 x-max = 20 x-scl = 1 y-min = 1880000 y-max = 2500000 y-scl = 5000 Observe these two models together and draw a rough sketch of what you see on the next page. Don't worry about labeling units or being extremely precise. Label the demand and supply functions. e.) Based on what you observe with these two graphs, do you think market equilibrium in the nursing field will be reached in the upcoming years? Explain your reasoning. Math\t1324\tLab\t1\t-\tChapter\t1\t&\t2 Page\t6