An automobile manufacturer sells cars in America, Europe, and Asia, charging a different price in each of the three markets. The price function for cars sold in America is p = 23-0.2x (for 0 ≤ x ≤ 115), the price function for cars sold in Europe is q = 11 -0.1y (for 0 ≤ y ≤ 110), and the price function for cars sold in Asia is r = 14 -0.1z (for 0 ≤z ≤ 140), all in thousands of dollars, where x, y, and z are the numbers of cars sold in America, Europe, and Asia, respectively. The company's cost function is C = 27 + 9(x+y+z) thousand dollars. (a) Find the company's profit function P(x, y, z). [Hint: The profit will be revenue from America plus revenue from Europe plus revenue from Asia minus costs, where each revenue is price times quantity.] P(x, y, z) = (b) Find how many cars should be sold in each market to maximize profit. [Hint: Set the three partials Px, Py, and P₂ equal to zero and solve. Assuming that the maximum exists, it must occur at this point.] America Europe Asia cars cars cars
An automobile manufacturer sells cars in America, Europe, and Asia, charging a different price in each of the three markets. The price function for cars sold in America is p = 23-0.2x (for 0 ≤ x ≤ 115), the price function for cars sold in Europe is q = 11 -0.1y (for 0 ≤ y ≤ 110), and the price function for cars sold in Asia is r = 14 -0.1z (for 0 ≤z ≤ 140), all in thousands of dollars, where x, y, and z are the numbers of cars sold in America, Europe, and Asia, respectively. The company's cost function is C = 27 + 9(x+y+z) thousand dollars. (a) Find the company's profit function P(x, y, z). [Hint: The profit will be revenue from America plus revenue from Europe plus revenue from Asia minus costs, where each revenue is price times quantity.] P(x, y, z) = (b) Find how many cars should be sold in each market to maximize profit. [Hint: Set the three partials Px, Py, and P₂ equal to zero and solve. Assuming that the maximum exists, it must occur at this point.] America Europe Asia cars cars cars
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![An automobile manufacturer sells cars in America, Europe, and Asia, charging a different price in each of the three markets. The price function for cars sold in America is p = 23 – 0.2x (for 0 ≤ x ≤ 115),
the price function for cars sold in Europe is q = 11 – 0.1y (for 0 ≤ y ≤ 110), and the price function for cars sold in Asia is r = 14 – 0.1z (for 0 ≤ z ≤ 140), all in thousands of dollars, where x, y, and z
are the numbers of cars sold in America, Europe, and Asia, respectively. The company's cost function is C = 27 + 9(x + y + z) thousand dollars.
(a) Find the company's profit function P(x, y, z). [Hint: The profit will be revenue from America plus revenue from Europe plus revenue from Asia minus costs, where each revenue is price times
quantity.]
P(x, y, z) =
(b) Find how many cars should be sold in each market to maximize profit. [Hint: Set the three partials Px, Py, and P₂ equal to zero and solve. Assuming that the maximum exists, it must occur at
this point.]
America
Europe
Asia
cars
cars
cars](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47741e47-318e-429d-9fcb-8d6f6d53bc4b%2F1127f562-f0a4-4913-99d6-c57ca8a87bdc%2Fgob0r0f_processed.png&w=3840&q=75)
Transcribed Image Text:An automobile manufacturer sells cars in America, Europe, and Asia, charging a different price in each of the three markets. The price function for cars sold in America is p = 23 – 0.2x (for 0 ≤ x ≤ 115),
the price function for cars sold in Europe is q = 11 – 0.1y (for 0 ≤ y ≤ 110), and the price function for cars sold in Asia is r = 14 – 0.1z (for 0 ≤ z ≤ 140), all in thousands of dollars, where x, y, and z
are the numbers of cars sold in America, Europe, and Asia, respectively. The company's cost function is C = 27 + 9(x + y + z) thousand dollars.
(a) Find the company's profit function P(x, y, z). [Hint: The profit will be revenue from America plus revenue from Europe plus revenue from Asia minus costs, where each revenue is price times
quantity.]
P(x, y, z) =
(b) Find how many cars should be sold in each market to maximize profit. [Hint: Set the three partials Px, Py, and P₂ equal to zero and solve. Assuming that the maximum exists, it must occur at
this point.]
America
Europe
Asia
cars
cars
cars
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