Suppose the average costs of a mining operation depend on the number of machines

Question

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by C(x) = 2500x + 1, 822, 500/x, x > 0, where x is the number of machines used. Find the critical values of C(x) that lie in the domain of the problem. (Enter your answers as a comma-separated list.) x = _________ Over what interval in the domain do average costs decrease? (Enter your answer using interval notation.) __________ Over what interval in the domain do average costs increase? (Enter your answer using interval notation.) _____________ How many machines give minimum average costs? _________ machines What is the minimum average cost? $ ____________

Explanation / Answer

In order to find the critical value; C'(x)=0

C(x) = 2500x + 1822500 x^(-1)
C'(x) = 2500 - 1822500 x^(-2)
Applying the condition C'(x) = 0

2500 - 1822500 x^(-2) = 0
2500 = 1822500 x^(-2)
x^2 = 1822500 / 2500
x^2 = 729
x = +/- 27, but since x > 0, therefore x = 27

Minimum average cost can be obtained at

put x=27 in C(x) = 2500x + 1822500 x^(-1)

c(27) = 2500 *27 + 1822500/27

= 135000

Now let's choose any other value of x, assume it as 26

C(26) = 2500 *26 + 1822500/26

= 135096.15 (which is greater than 135000)

hence min value is obtained at x= 27 and min cost is 135000

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