Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Total Revenue and Profit This is a continuation of Exercise 15. The total revenue
The profit
Suppose the manufacturer of widgets in Exercise 15 sells the widgets for
a. Use a formula to express this manufacturer’s total revenue
b. Use a formula to express the profit
c. Express using functional notation the profit of this manufacturer if there are
d. At the production level of
Suppose that a manufacturer of widgets has fixed costs of
a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use.
b. Express using functional notation the total cost if there are
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- Reminder Round all answer to two decimal places unless otherwise indicated. Total Revenue and Profit This is a continuation of Exercise 13. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. In this exercise, we assume that the selling price per unit of the item is a constant, so it does not depend on the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If the profit is zero, then the manufacturer is at a break-even point. We consider again the manufacturer of widgets in Exercise 13 with fixed costs of 1500 pr month and a variable cost of 20 per widget. Suppose the manufacturer sells 100 widgets for 2300 total. a. Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as function of the number N of widgets produced in a month. b. Use a formula to express the monthly profit P, in dollars, of this manufacturer as function of the number of widgets produced in a month. Explain how the slope and initial of P are derived from the fixed costs, variable cost, and price per widget. c. What is the break-even point for this manufacturer? d. Make graphs of total monthly cost and total monthly revenue. Include monthly production levels up to 1200 widgets. What is the significance of the point where the graphs cross?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Profit The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. In this exercise, we measure all monetary values in dollars. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering such things as plant maintenance and insurance as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. The total revenue R for a manufacturer during a given time period is also a function of the number N of items produced during that period. To determine a formula for the total monthly revenue, we need to know the selling price p per unit of the item, which in general is also a function of N. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. Suppose that a manufacturer of widgets has a fixed costs of 2000 per month and that the variable cost is 30 per widget. Further, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. Number N Price p 200 41.00 250 40.50 300 40.00 350 39.50 a.Use a formula to express the total monthly cost C of this manufacturer as a function of N. b.Use the table to find a linear model of p as a function of N. c.Use your answer to part b to find a formula expressing the total monthly revenue R as function of N. d.Use your answers to part a and part c to find a formula expressing the monthly profit P as a function of N. What type of function is the profit: linear or quadratic? e.Find the two monthly production levels at which the manufacturer just breaks even that is, where the profit is zero.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. More on ProfitThis is a continuation of Exercises 15, 16, and 17. In this exercise, we use the formula for the total cost of the widget manufacturer found in Exercise 15 and the formula for the total revenue found in Exercise 17. a.Use a formula to express the profit P of this manufacturer as a function of N. b.Consider the three production levels: N=200, N=700, and N=1200 . For each of these, determine whether the manufacturer has a loss, turns a profit, or is a break even point. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value. 16.Total Revenue and ProfitThis is a continuation of Exercise 15. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at break-even point. Suppose the manufacturer of widgets in Exercise 15 sells the widgets for 25each. a.Use a formula to express this manufacturers total revenue R in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b.Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use. c.Express using functional notation the profit of this manufacturer if there are 250 widgets produced in a month, and then calculate that value. d.At the production level of 250 widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of 1000 widgets per month? 17.More on RevenueThis is a continuation of Exercise 15 and 16. In general, the highest price p per unit of an item which a manufacturer can sell N items is not constant, but is rather a function of N. The total revenue R is still the product of p and N, but the formula for R is more complicated when p depends on N. Suppose the manufacturer of widgets in Exercises 15 and Exercises 16 no longer sells widgets for 25 each. Rather, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. a.Verify that the formula p=500.01N where p is the price in dollars, give the same values as those in the table. N=Numberofwidgetssold p=Price 100 49 200 48 300 47 400 46 500 45 b.Use the formula from part a and the fact that R is the product of p and N to find a formula expressing the total revenue R as a function of N for this widget manufacturer. c.Express using functional notation the total revenue of this manufacturer if there are 450 weights produced in a month, and then calculate that value.arrow_forward
- Reminder: Round all answer to two decimal places unless otherwise indicated. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To deter mine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost C of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know two things. The first is the manufacturers fixed costs. This amount covers expenses such as plant maintenance and insurance, and it is the same no matter how many items are produced. The second thing we need to know is the cost for each unit produced, which is called the Variable cost. Suppose that a manufacturer of widgets has fixed costs of 1500 per month and that the variable cost is 20 per widget so it costs 20 to produce 1 widget. a. Explain why the function giving the total monthly cost C, in dollars, of this widget manufacturer in terms of the number N of widgets produced in a month is linear. Identify the slope and initial value of this function, and write down a formula. b. Another widget manufacturer has a variable cost of 12 per widget, and the total is 3100 when 150 widgets are produced in a month. What are the fixed costs for this manufacturer? c. Yet another widget manufacturer has determined the following: The total cost is 2700 when 100 widgets are produced in a month, and the total cost is 3500 when 150 widgets are produced in a month. What are the fixed costs and variable costs for this manufacturer?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Cost and Revenue The cost C and the revenue R for a brokerage firm depend on the number T of transactions executed. Both C and R are measured in dollars. It costs 750 per day to keep the office open, and brokers are paid an average of 25 per transaction. Also, 35 in fees are collected for each transaction. a. Find a formula that gives C as a function of T. b. Find a formula that gives R as a function of T. c. Find the number of daily transactions that are needed to make the revenue 1500 more than the cost.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. More on RevenueThis is a continuation of Exercises 15 and 16. In general, the highest price p per unit of an item which a manufacturer can sell N items is not constant, but is rather a function of N. The total revenue R is still the product of p and N, but the formula for R is more complicated when p depends on N. Suppose the manufacturer of widgets in Exercises 15 and Exercises 16 no longer sells widgets for 25 each. Rather, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. a.Verify that the formula p=500.01N where p is the price in dollars, give the same values as those in the table. N=Numberofwidgetssold p=Price 100 49 200 48 300 47 400 46 500 45 b.Use the formula from part a and the fact that R is the product of p and N to find a formula expressing the total revenue R as a function of N for this widget manufacturer. c.Express using functional notation the total revenue of this manufacturer if there are 450 weights produced in a month, and then calculate that value. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value. 16.Total Revenue and ProfitThis is a continuation of Exercise 15. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at break-even point. Suppose the manufacturer of widgets in Exercise 15 sells the widgets for 25each. a.Use a formula to express this manufacturers total revenue R in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b.Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use. c.Express using functional notation the profit of this manufacturer if there are 250 widgets produced in a month, and then calculate that value. d.At the production level of 250 widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of 1000 widgets per month?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. A wedding Reception You rent a wedding venue for a cost of 3200. The cost includes a catered lunch for 50 guests. For each addition guest, though, the catered lunch costs 31. a. What is the cost of the venue and lunch if you invite 100 guests. b. Find a formula showing the cost of the venue and the lunch as a function of n, the number of guests. Assume that n is at least 50. c. The amount you have budgeted for the venue and catered lunch is 5500. How many guests can you invite.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Math and the City An article in The New York Times states, "The number of gas stations in a city grows only in proportion to the 0.77 power of population. This means that the approximate number G of gas stations in a city is a power function of the population N, and the power is k=0.77. That is, G=cN0.77, where c is some as yet unknown constant. We measure N in millions. a. If one city is twice as large as another, how do the numbers of gas stations compare? b. The population of Houston, Texas, is 2.2million and, according to Yahoo Local, there are 1239 gas stations in Houston. Use this information to find the value of c. c. Los Angeles has a population of about 3.9million. Using the value of c that you found in part b, estimate the number of gas stations in Los Angeles. Round your answer to the nearest whole number. Note: According to Yahoo Local, the correct number is 2013.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Kleibers law states that for the vast majority of animals, the metabolic rate M is a power function of the weight W, and the power is k=34. a.The Eastern gray squirrel weighs about 1 pound. How does the squirrels metabolic rate compare with that of a 200 -pound man? b.How does a 200- pound mans metabolic rate compare with that of a 130- pound woman? c.Based on your answer to part b, would overeating the same amount for each be more likely to lead to weight gain for the man or for the woman?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. 12. A Car That Gets 32 Miles per Gallon The cost C of operating a certain car that gets 32 miles per gallon is a function of the price g, in dollars per gallon, of gasoline and the distance d, in miles, that you drive. The formula for C=C(g,d) is C=gd/32 dollars. a. Use functional notation to express the cost of operation if gasoline costs 98 cents per gallon and you drive 230 miles. Calculate the cost. b. Calculate C(3.53,172) and explain the meaning of the number you have calculated.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Sales Growth A study of the sales s, in thousands of dollars, of a product as a function of time t, in years, yields the equation of change dsdt=0.3s(4s). This is valid for s less than 5. a.What level of sales will be attained in the long run? b.What is the largest rate of growth in sales?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning