Word Problem

Question

A medical equipment industry manufactures X-ray machines. The unit cost C (the cost in dollars to make each X-ray machine) depends on the number of machines made. If machines are made, then the unit cost is given by the function 𝐶(𝑥) = 0.4𝑥2 − 224𝑥 + 46,529. What is the minimum unit cost? Do not round your answer.

Answer 1

unit cost = C(x) divided by x = C(x)/x = 0.4x -224 + 46,529/x = Average Total Cost per unit = ATC

for Average Variable Cost ignore the fixed cost and AVC = 0.4x -224

to get minimum ATC set C(x)/x = zero and solve for x, the plug that value into C(x)/x

C(x)/x = 0.4x -224 + FC/x = 0 where FC = fixed cost = 46,529

multiply by x, which gives C(x) = 0, a quadratic equation solved by the quadratic formula

or completing the square

An easier method is notice that MC = ATC at minimum ATC

MC = marginal cost = derivative of the Cost function

C'(x) = 0.8x -224 = ATC = 0.4x -224 +46,529/x

which gives

0.4x = 46,529/x

multiply both sides by x

04x^2 = 46,529

x^2 = 46,529/0.4 = 465,290/4 = 116,322.5

x = sqr116322.5 = 341.06 to 2 decimal places