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?

x = 8 years

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

Step-by-step 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