Question

The rate at which an assembly line workers efficiency E (expressed as a percent) changes with respect to time t is given by E'(t) = 75 - 6t, where t is the number of hours since the workers shift began. Assuming that E(1) = 92, find E(t).

Answer 1

By the fundamental theorem of calculus,

$E(t)=E(1)+\displaystyle\int_1^t E'(u)\,\mathrm du$

So we have

$E(t)=92+\displaystyle\int_1^t(75-6u)\,\mathrm du$

$E(t)=92+(75u-3u^2)\bigg|_1^t$

$E(t)=20 + 75 t - 3 t^2$