C(x)=75[squareroot(x) -10]. (C is cost) what is marginal cost at 1225...?

Question

how much production is needed to achieve a marginal cost of $0.50? i think i know how to do it. i just want to double check with some you you guys

Answer 1

So, the marginal cost is just the first derivative with respect to x.
I'm going along with just the x being inside the square root and not the 10. To help things out you might go ahead and distribute the 75.

C(x) = 75sqrt(x) - 750 = 75x^(1/2) - 750

Taking the derivative isn't too bad here,
C'(x) = 75*(1/2)x^(-1/2).
Plug in 1225 for x and you get 75*(1/2)/35 or 75/70, which in turn reduces to 15/14 or 1 1/14.

If you want the marginal cost to equal $0.50 just set C' = to .5 and solve for x. 75*(1/2)x^(-1/2) = 1/2. So, 75 = x^(1/2). Which makes x = 75^2 = 5625