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(2) The demand function for a certain product is linear and defined by the equation p(r) = 10 – , where r is the total output. Find the level of production at which the company has the maximum revenue.

(2) The demand function for a certain product is linear and defined by the equation p(r) = 10 – , where r is the total output. Find the level of production at which the company has the maximum revenue.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(1) A game console manufacturer determines that in order to sell x units, the price per one unit (in dollars) must decrease by the demand function given by p(x) = 500- 0.1r, where p is measured in dollars/device. The manufacturer also determines that the cost depends on the volume of production and includes a fixed part, 100,000 (dollars), and a variable part, 100x (dollars), which gives us the cost function C(x) = 100000 + 100x. (a) What price per unit must be charged to get the maximum profit? How many game consoles must be sold to achieve the maximum profit? (Hint: Profit is determined by subtracting costs from revenue, and you need to determine the revenue function.) (b) Find the marginal profit at a = 100 units and interpret the meaning of the solution in context (Hint: The derivative of a function is called the marginal function.) (2) The demand function for a certain product is linear and defined by the equation p(x) = 10 – 5, where x is the total output. Find the level of production at which the company has the maximum revenue. 2 (3) A diesel generator burns fuel at the rate of G(x) = (338 + 2x) liters per hour when producing a thousand kilowatt hours of electricity. Suppose that fuel costs 60 cents a liter. Find the value of r that leads to minimum total cost if the generator is operated for 32 hours. 12 (4) To produce r units of some product a company spends C(x) = ax? + bx dollars, where a and b are positive real numbers. The product is sold at price p dollars per unit. Determine the sales volume at which profit reaches its maximum. (5) The production cost per a period of time is given by the quadratic function C(r) = a + bx², where a and b are some positive real numbers and x represents the number of units. Find the minimal average cost. (Hint: The average cost is the total cost divided by the number of units produced.) %3D (6) A manufacturing company needs to build a box whose base length is 6 times the base width and they must ensure that the box will enclose 20 in of volume. The cost of the material of the sides is $3/in² and the cost of the top and bottom is $15/in?. Determine the dimensions of the box that will minimize the cost. (7) CHALLENGE PROBLEM: A company sells wholesale lots of goods at a price of $20 per thousand for orders of 10,000 goods or less, with the charge per thousand decreased by 50 cents for each thousand above 10,000 goods. Determine the order size at which the company has the maximum revenue. (Hint: The revenue function is a piecewise-defined function Let the independent variable be the order size in thousands of goods ....doc W- Review for the f....docx MacBook Pro 2.
Transcribed Image Text:(1) A game console manufacturer determines that in order to sell x units, the price per one unit (in dollars) must decrease by the demand function given by p(x) = 500- 0.1r, where p is measured in dollars/device. The manufacturer also determines that the cost depends on the volume of production and includes a fixed part, 100,000 (dollars), and a variable part, 100x (dollars), which gives us the cost function C(x) = 100000 + 100x. (a) What price per unit must be charged to get the maximum profit? How many game consoles must be sold to achieve the maximum profit? (Hint: Profit is determined by subtracting costs from revenue, and you need to determine the revenue function.) (b) Find the marginal profit at a = 100 units and interpret the meaning of the solution in context (Hint: The derivative of a function is called the marginal function.) (2) The demand function for a certain product is linear and defined by the equation p(x) = 10 – 5, where x is the total output. Find the level of production at which the company has the maximum revenue. 2 (3) A diesel generator burns fuel at the rate of G(x) = (338 + 2x) liters per hour when producing a thousand kilowatt hours of electricity. Suppose that fuel costs 60 cents a liter. Find the value of r that leads to minimum total cost if the generator is operated for 32 hours. 12 (4) To produce r units of some product a company spends C(x) = ax? + bx dollars, where a and b are positive real numbers. The product is sold at price p dollars per unit. Determine the sales volume at which profit reaches its maximum. (5) The production cost per a period of time is given by the quadratic function C(r) = a + bx², where a and b are some positive real numbers and x represents the number of units. Find the minimal average cost. (Hint: The average cost is the total cost divided by the number of units produced.) %3D (6) A manufacturing company needs to build a box whose base length is 6 times the base width and they must ensure that the box will enclose 20 in of volume. The cost of the material of the sides is $3/in² and the cost of the top and bottom is $15/in?. Determine the dimensions of the box that will minimize the cost. (7) CHALLENGE PROBLEM: A company sells wholesale lots of goods at a price of $20 per thousand for orders of 10,000 goods or less, with the charge per thousand decreased by 50 cents for each thousand above 10,000 goods. Determine the order size at which the company has the maximum revenue. (Hint: The revenue function is a piecewise-defined function Let the independent variable be the order size in thousands of goods ....doc W- Review for the f....docx MacBook Pro 2.
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