Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar.
A)$0
B)$35
C)$490
D)$245
E)$8,582

Question

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by  ,  , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. ​
A)$0 
B)$35 
C)$490 
D)$245 
E)$8,582

Quizplus · Accepted Answer

To find the minimum average cost, we need to find the minimum point of the function. To do so, we take the derivative of the function and set it equal to 0.

d/dx (420x + 10000/x) = 420 - 10000/x^2

420 - 10000/x^2 = 0
x^2 = 10000/420
x = sqrt(100/42) = 3.3 (rounded to the nearest tenth)

Since we cannot use a fractional number of machines, we round up to 4 machines. Therefore, the minimum average cost is:

420(4) + 10000/4 = $490 (rounded to the nearest dollar).

Thus, the answer is C.

Suppose the Average Costs of a Mining Operation Depend on the Number