Question

Suppose a company wants to introduce a new machine that will produce a marginal annual savings in dollars given by S ?(x)=(125/2)?x^2?,

where x is the number of years of operation of the? machine, while producing marginal annual costs in dollars of C ?(x)=x^2+(5/2)x.

a. To maximize its net? savings, for how many years should the company use this new? machine?

b. What are the net savings during the first year of use of the? machine?

c. What are the net savings over the period determined in part? a?

Answer 1

Given, machine will produce a marginal annual savings in dollars given by S(x)=(125/2)x² and marginal annual cost for machine in dollars given by C(x)=x²+(5/2)x.

a) By given condition we have,

[x²-(x-1)²]+(5/2)[x-(x-1)] > (125/2)[x²-(x-1)²]

i.e., (2x-1)+5/2 > (125/2)(2x-1)

i.e., 123x < 64

which is impossible.

To maximize net savings, the company should use this new machine forever until it damaged.

b) For 1st year :

Marginal annual savings : S(1) = $[(125/2)*1²] = $[(125/2)*1] = $62.5

Marginal annual costs : C(1) = $[1²+(5/2)*1] = $[1+(5/2)] = $3.5

Therefore, net savings = $(62.5-3.5) = $59.

c) Here, net savings is uncountable.