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 50 th 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 \( C(x)=x^{2}-140 x+8800 \). A. How many players should be produced to minimize the marginal cost? B. What is the minimum marginal cost? A. To minimize the marginal cost, \( \square \) thousand mp3 players should be produced. B. The minimum marginal cost is \( \$ \square \).
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To minimize the marginal cost, we first need to find the derivative of the cost function \( C(x) \) and set it to zero. The function is \( C(x) = x^2 - 140x + 8800 \). Taking the derivative, we get \( C'(x) = 2x - 140 \). Setting \( C'(x) = 0 \) gives us \( 2x - 140 = 0 \), or \( x = 70 \). Therefore, to minimize the marginal cost, **70 thousand mp3 players** should be produced. To find the minimum marginal cost, we substitute \( x = 70 \) back into the original cost function. We calculate \( C(70) = 70^2 - 140(70) + 8800 = 4900 - 9800 + 8800 = 4900 \). Therefore, the minimum marginal cost is **\$4900**.
