Math 210 Spring 2000 Exam # 1A1.) You and some friends have decided to start your own record label, Sundevil Discs. The production costs of your first compact disk (CD) are $2.14 per CD plus $20,000 for studio recording time. You plan to sell the CDs for $15.99 each.a) [3 pts.]Find a formula for C(x), the total cost (in dollars) as a function of x, the number of CDs produced.Answer: 000,2014.2)(  xxCb.) [3 pts.]Find a formula for R(x), the total revenue for selling x CDs.Answer: xxR 99.15)( c.) [3 pts.]Assuming you sell every CD you produce, how many CDs would you need to sell to make $10,000 in profit? Show your work.Answer: 2167CDs of #06.2166000,10000,2014.299.15xxx2.) The table shows the number of companies having world-wide web sites in a certain region from 1991 to 1997.Year 1991 1992 1993 1994 1995 1996 1997Companies 650 1900 5600 16,600 48,900 144,000 423,700a.) [3 pts.]Letting t = 0 correspond to 1991, determine the best type of model and give theequation. Explain why the type you chose is the best (use complete sentences).Answer: ExponentialxtN )947.2(273.647)( Choice made from: either from exponential growth curvature of stat-plot and curve-fit,or from calculating the percentage rate of growth which approximately 1.94.b.) [3 pts.]According to the model, when will there be 5,000,000 companies with web sites?Answer: 28.8 )947.2ln(/)273.647/000,000,5ln( )947.2(273.647000,000,5xx In year 2000c.) [3 pts.]Would this be a good model to forecast the number of companies with web sites in the year 2050. Explain why or why not (use complete sentences)?Answer: No (multiple answers for reasons!)3.) Larry Liebnitz is running for mayor of Newtonville. The total percentage of the voters that will vote for Larry is dependent on the amount of money he spends campaigning. This relationship can be modeled byd.e.)d(P05051564where d is the amount spent in hundreds of dollars and P(d) is the percentage of the vote.a. [3 pts.]Sketch a graph of the model on the given axes and label the axes.b. [3 pts.]According to the model, what is the limiting value of the percentage of the voters who vote for Larry Liebnitz? Explain your answer (use complete sentences).Answer: 64.5%4.) The table shows the monthly profit of an amusement park for various ticket prices.Ticket price (dollars) 20 22 24 26 28 30 32Profit (millions of dollars) 14.4 19.4 23.1 25.2 26.0 25.2 23a.) [4 pts.] Which of the following types of models is the best for this data? Explain why it is the best. (cubic, exponential, linear, logistic, quadratic)Answer: either from parabolic curvature of stat-plot and curve-fit, or from calculating the second differences which is approximately -1.4.b.) [3 pts.] Find the best model for the data.Answer: 4.1161756.101819.)(2 xxxP (Rounded model)c.) [4 pts.] According to the model found in part b.), at what ticket price will the amusement park begin to post a net loss (i.e, negative profit)?Answer: $39.935.) The following function gives the profit of a company in thousands of dollars in terms of the number of units produced.13651202 x.x.)x(Pa) [4 pts.] Find the average rate of change of the profit over the first 20 units.Answer: Average rate of change of the profit from zero production to 20 units production = 96.22012.58020)0()20( PP thousand dollars per unitb) [4 pts.] Will the instantaneous rate of change of the profit be greater at 15 units or at 30 units? Explain (use complete sentences). Answer: At x = 15 the slope is positive and at x = 30 the slope is negative. Hence, the instantaneous rate of change of the profit be greater at 15 units.c.) [5 pts.]In terms of the profit, what would a negative instantaneous rate of change mean (use complete sentences)?Answer: It means that the profit is decreasing.d.) [6 pts.] What are the units for dxdP? thousand dollars per unitGiven that96210 .)(P , explain, in the context of the problem what this means.Answer: When the number of units produced is 10, the profit is increasing at the rate of 2.96 thousand dollars per unit, i.e. with the production of one additional unit from 10 units, the profit would go up by 2.96 thousand dollars.6.) [8 pts.] Let V(m) give the volume of fructose syrup in gallons held by a tank m minutes after the first shift begins working at a soft drink bottling plant.Given the following facts, sketch a possible graph for V(m) labeling the axes.I.gallons)(V 5500 II.utemin/gallons)('V 1015 III.utemin/gallons)('V 030 IV.utemin/gallons)('V 2045 V. There is no inflection point.Answer: The following graph is one of the possibilities7.) The graph below represents the function )t(G, your score on the next MAT 210 test after you have studied t hours during the two days before the test. a. [3 pts.]Draw the secant line whose slope gives the average rate of change of the score as a student increases the number of hours of study from 0 hours to 4 hours.(see graph)b. [4 pts.]Find the average rate of change of the score as a student increases the number of hours of study from 0 hours to 4 hours.Answer: hourper points 3412046072 c. [3 pts.]Draw the tangent line whose slope represents instantaneous rate of change of the score at 8 hours of study. (see graph)d. [5 pts.]Estimate the instantaneous rate of change of the score at 8 hours of study.Answer: hourper points 5.1245061012070Score out of 100 Points0204060801001200 2 4 6 8 10t hours of studyScore8.) [12 pts.]On the following graph points B and C are points of inflection. Use the graphto complete the following table.Graphat PointConcavity (up, down,neither)Slope of tangent line (positive, negative, neither)Tangent line (below, above or through curve)A up negative belowB neither positive throughC neither negative throughD up positive