Question

Suppose a company wants to introduce a new machine that will produce a rate of annual savings​ (in dollars) given by the functionS'(x)​, where x is the number of years of operation of the​ machine, while producing a rate of annual costs​ (in dollars) given by the function C'(x).

S'(x) = 142 - x², C'(x) = x² + (7/4) x
For how many years will it be profitable to use this new​ machine?

Answer 1

x = 8 years

Therefore, the machine will be profitable to use for 8 years

Explanation:

Given;

The rate of annual savings​ (in dollars) given by the function S'(x)​, and the rate of annual costs​ (in dollars) given by the function C'(x).

S'(x) = 142 - x²

C'(x) = x² + (7/4) x

For the machine to be profitable for use, The rate of annual savings​ (in dollars) given by the function S'(x)​ equal to or greater than the rate of annual costs​ (in dollars) given by the function C'(x).

S'(x) = C'(x)

142 - x² = x² + (7/4) x

0 = 2x² + (7/4)x - 142

2x² + (7/4)x - 142 = 0

Multiplying through by 4;

8x² + 7x - 568 = 0

Solving the quadratic equation, we have;

x = -8.875 or x = 8

Since x which is the number of years cannot be negative.

x = 8 years

Therefore, the machine will be profitable to use for 8 years