Concept explainers
Total Cost The background for this exercise can be found in Exercises 13 and 14 in Section 3.2. The following table gives the total cost C, in dollars, for a widget manufacturer as a function of the number N of widgets produced during a month.
Number N | Total cost C |
200 | 7900 |
250 | 9650 |
300 | 11, 400 |
350 | 13, 150 |
a. What are the fixed costs and variable cost for this manufacturer?
b. The manufacturer wants to reduce the fixed costs so that the total cost at a monthly production level of 350 will be $12, 975. What will the new fixed costs be?
c. Instead of reducing the fixed costs as in part b, the manufacturer wants to reduce the variable cost so that the total cost at a monthly production level of 350 will be $12, 975. What will the new variable cost be?
Trending nowThis is a popular solution!
Chapter 3 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Tuition at American Public Universities This is a continuation of Exercise 6. The following table shows the average yearly in-state tuition and required fees, in dollars, charged by four-year American public universities in the school year ending in the given year. Date Average tuition 2012 8318 2013 8595 2014 8872 2015 9149 2016 9426 a. Show that these data can be modeled by a linear function, and find its formula. b. What is the slope for the linear function modeling tuition and required fees for public universities? c. What is the slope of the linear function modeling tuition and required fees for private universities? Note: See Exercise 6. d. Explain what the information in parts b and c tells you about the rate of increase in tuition in public versus private institutions. e. Which type of institution shows the larger percentage increase from 2015 to 2016? 6. Tuition at American Private Universities The following table shows the average yearly tuition and required fees, in dollars, charged by four-year American private nonprofit universities in the school year ending in the given year. Date Average tuition 2012 27, 870 2013 29, 004 2014 30, 138 2015 31, 272 2016 32, 406 a. Show that these data can be modeled by a linear function, and find its formula. b. Plot the data points and add the graph of the linear formula you found in part a. c. What prediction does this formula give for average tuition and fees at four-year American private nonprofit universities for the academic year ending in 2021?arrow_forwardBreaking Even The background for this exercise can be found in Exercises 15, 16, 17, and 18 in Section 1.4. A manufacturer of widgets has fixed costs of 700 per month, and the variable cost is 65 per thousand widgets so it costs 65 to produce 1 thousand widgets. Let N be the number, in thousands, of widgets produced in a month Find a formula for the manufacturers total cost C as a function of N. a. Find a formula for the manufacturers total cost C as a function of N. b. The highest price p, in dollars per thousand widgets, at which N can be sold is given by the formula p=750.02N. Using this, find a formula for the total revenue R as a function of N. c. Use your answers to parts a and b to find a formula for the profit p of this manufacturer as a function of N. d. Use your formula from part c to determine the two break-even points for this manufacturer Assume that the manufacturer can produce at most 500 thousand widgets in a month.arrow_forwardHigh School Graduates The following table shows the number, in millions, graduating from high school in the United States in the given year. Year Number graduating in millions 1985 2.83 1987 2.65 1989 2.47 1991 2.29 a. By calculating difference, show that these data can be modeled using a linear function. b. What is the slope for the linear function modeling high school graduations? Explain in practical terms the meaning of the slope. c. Find a formula for a linear function that models these data. d. Express, using functional notation, the number graduating from high school in 1994, and then use your formula from part c to calculate that value.arrow_forward
- Average Cost The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6000. The inventor sells each game for $1.69. Let x be the number of games produced. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost C as a function of the number of games produced. (b) Write the average cost per unit C=Cx as a function of x.arrow_forwardBusiness The annual cost C (in thousands of dollars) and revenue R (in thousands of dollars) for a company each year from 2010 through 2016 can be approximated by the models C=2549t+1.1t2andR=341+3.2t where P is the year, with t=10 corresponding to 2010. (a) Write a function P that represents the annual profit of the company. (b) Use a graphing utility to graph C,R, and P in the same viewing window.arrow_forwardLater High School Graduates This is a continuation of Exercise 16. The following table shows the number, in millions, graduating from high school in the United States in the given year. Year Number graduating in millions 2001 2.85 2003 2.98 2005 3.11 2007 3.24 a. Find the slope of the linear function modeling high school graduations, and explain in practical terms the meaning of the slope. b. Find a formula for a linear function that models these data. c. Express, using functional notation, the number graduating from high school in 2008, and then calculate the value. d. The actual number graduating from high school in 1994 was about 2.52 million. Compare this with the value given by the formula in part b and with your answer to part of Exercise 16. Which is closer to the actual value? In general terms, what was the trend in high school graduations from 1985 to 2007? 16. High School Graduates The following table shows the number, in millions, graduating from high school in the United States in the given year.16 Year Number graduating in millions 1985 2.83 1987 2.65 1989 2.47 1991 2.29 a. By calculating difference, show that these data can be modeled using a linear function. b. What is the slope for the linear function modeling high school graduations? Explain in practical terms the meaning of the slope. c. Find a formula for a linear function that models these data. d. Express, using functional notation, the number graduating from high school in 1994, and then use your formula from part c to calculate that value.arrow_forward
- Traffic Accidents The following table shows the cost C of traffic accidents. in cents per vehicle-mile, as a function of vehicular speed s, in miles per hour, for commercial vehicles driving at night on urban streets. Speed s 20 25 30 35 40 45 50 Cost C 1.3 0.4 0.1 0.3 0.9 2.2 5.8 The rate of vehicular involvement in traffic accidents per vehicle-mile can be modeled as a quadratic function of vehicular speed s, and the cost per vehicular involvement is roughly a linear function of s, so we expect that C the product of these two functions can be modeled as a cubic function of s. a. Use regression to find a cubic model for the data. Keep two decimal places for the regression parameters written in scientific notation. b. Calculate C(42) and explain what your answer means in practical terms. c. At what speed is the cost of traffic accidents for commercial vehicles driving at night on urban streets at a minimum? Consider speeds between 20 and 50 miles per hour.arrow_forwardTEST YOUR UNDERSTADING FOR EXAMPLE 3.3 The State of New York also has an income tax. The tax table for New York shows that a single New York resident with a taxable income of 15,000 owes 671 in New York income tax. If the taxable income is 15,500, then the table shows a tax liability of 701. Assume that the state tax is a linear function of taxable income, and find the tax owed if the taxable income is 15,350. EXAMPLE 3.3 OKLAHOMA INCOME TAX The amount of income tax T=T(l), in dollars, owed to the state of Oklahoma is a linear function of the taxable income l, in dollars, at least over a suitably restricted range of incomes. According to the Oklahoma Income Tax table for the year 2015, a single Oklahoma resident taxpayer with a taxable income of 15,000 owes 579 in Oklahoma income tax. In functional notation, this is T(15,000)=579. If the taxable income is 15,500, then the table shows a tax liability of 605. Part 1 Calculate the rate of change in T with respect to I, and explain in practical terms what it means. Part 2 How much does the taxpayer owe if the taxable income is 15,350?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning