marginal cost of a product

Question

The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For​ example, if the marginal cost of producing the 50th product is​ $6.20, it cost​ $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C​ (in dollars) to produce x thousand mp3 players is given by the function Upper C left parenthesis x right parenthesis equals x squared minus 140 x plus 8300.C(x) = x2−140x+8300.

A. How many players should be produced to minimize the marginal​ cost?

B. What is the minimum marginal​ cost?

Answer 1

Hello Jacob,

The marginal cost function for producing x thousand mp3 players is given by

C(x) = x² - 140x + 8300.

a) For part a), you want to minimize the marginal cost function. That is, you want to find the value of x which minimizes the value of C(x). C(x) is a quadratic function, and in fact, its graph is an upward parabola. Therefore, the minimum value of C will occur at the vertex of the parabola. There are several ways to find the vertex, but the simplest way is probably to use the Vertex Theorem:

The vertex of the quadratic function C(x) = ax² + bx + c is given by (h,k), where h = -b/2a, and k = C(h). Thus, the x-coordinate of the vertex is

h = -b/2a = -(-140)/2(1) = 70.

The marginal cost is minimized when x = 70. (That is, when 70 thousand mp3 players are produced.)

b) The minimum marginal cost will be the y-coordinate of the vertex of the graph of C(x). (That is, the value of C(x) when x = 70.)

Min. Marginal Cost = C(h) = C(70)

= (70)² - 140(70) + 8300.

Min. Marginal Cost = $1300.

Hope that helps! Let me know if you have any more questions.

William