Answer
a) profitable number of years to use the machine $=8$ years
b) net total savings during first year $\approx 148\hspace{0.1cm}\text{dollars}$
c) net total savings over the entire period of use $\approx 771\hspace{0.1cm}\text{dollars}$
Work Step by Step
Rate of annual savings $S(x)=150-x^2$
Rate of annual costs $C(x)=x^2+\frac{11}{4}x$
$x:$ The number of years of operation of the machine
a) The machine can be used profitably as long as the rate of savings is more than the rate of costs. The maximum number of such years is given by the solution of,
$S'(x)=C'(x)$
$150-x^2=x^2+\frac{11}{4}x\implies 2x^2+\frac{11}{4}x-150=0$
Solving this gives $x=8$ or $x=-9.375$.
Since $x$ is number of years it cannot be negative. Hence $x=8$.
b) Net total savings during the first year is given by,
$\int_0^1 S'(x)-C'(x) dx$
$=\int_0^1 150-2x^2-\frac{11}{4}x$
$=147.958\approx 148\hspace{0.1cm} \text{dollars}$
c) Net total savings during the entire period of usage is given by,
$\int_0^8S'(x)-C'(x) dx$
$=\int_0^8 150-2x^2-\frac{11}{4}x$
$=770.666\approx 771\hspace{0.1cm} \text{dollars}$
