Question

6.(7 pts.) if the cost function for producing ( x ) items is given by ( c(x) = 500 + 2x + 0.1x^2 ) dollars, then the marginal cost at the production level of 50 unit is:
(a) 22 dollars
(b) 850 dollars per item
(c) 850 dollars
(d) 512 dollars per item
(e) 12 dollars per item.

Explanation:

Step1: Recall marginal cost formula

Marginal cost is the derivative of the cost function, so first find \( C'(x) \).
Given \( C(x) = 500 + 2x + 0.1x^2 \), the derivative \( C'(x) = 2 + 0.2x \) (using power rule: derivative of \( x^n \) is \( nx^{n - 1} \), derivative of constant is 0).

Step2: Evaluate at \( x = 50 \)

Substitute \( x = 50 \) into \( C'(x) \):
\( C'(50)=2 + 0.2\times50 \)
\( = 2 + 10 \)
\( = 12 \) dollars per item.

Answer:

(e) 12 dollars per item.