An automobile manufacturer sells cars in America, Europe, and Asia, charging a d

Question

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 = 26 0.2x (for 0 x 130), the price function for cars sold in Europe is q = 15 0.1y (for 0 y 150), and the price function for cars sold in Asia is r = 16 0.1z (for 0 z 160), 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 = 23 + 8(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 Pz

equal to zero and solve. Assuming that the maximum exists, it must occur at this point.]

Explanation / Answer

Profit = 26x – 0.2x2 +15y – 0.1y2 +16z -0.1z2 -23 -8x -8y -8z

P = 18x – 0.2x2 +7y – 0.1y2 +8z – 0.1z2 -23

x = 180/4 = 45

Py = 15-0.2y =0

y =150/2 = 75

Pz = 16- 0.2z =0

z = 16/20 = 80

America cars = 45

Europe cars = 75

Asia Cars = 80.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.