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 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.) 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? What is the minimum average cost?

Explanation / Answer

Critical values happen where the derivative is zero (or nonexistent/undefined).
C(x) = 2400x + 960,000x^(-1)
C'(x) = 2400 - 960,000x^(-2)
So where does C'(x) = 0?
0 = 2400 - 960,000x^(-2)
2400 = 960,000x^(-2)
x^2 = 960,000/2400
x^2 = 484
x = +/- 20, but since x > 0, therefore x = 20


If x = 20, C(x) = 2400(22) + (960,000/20) = 52,800 + 48,000 = 100,800
Choose any other value of x, and you'll find the C(x) > 100,800
For example if x = 19, C(x) =~ 119928.57
Therefore, to minimize C(x), you set x = 22 and get C(x) = 100800

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