Question

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector t hr into the 8 a.m. to 12 noon shift is approximately:
N'(t) = −3t^2 + 12t + 45 (0 <_ t <_4).
(a) Find an expression N(t) that approximates the number of sets inspected at the end of t hours.

Answer 1

Answer: $N(t)=-t^3+6t^2+45t$

Step-by-step explanation:

Since we have given that

$N'(t)=-3t^2+12t+45$

in between (0≤t≤4).

We need to find N(t) that approximates the number of sets inspected at the end of t hours.

So, it becomes,

$N(t)=\int\limits^4_0 {N'(t)} \, dt\\\\N(t)=\int\limits^4_0 {-3t^2+12t+45} \, dt\\\\N(t)=\dfrac{-3t^3}{3}+12\dfrac{t^2}{2}+45t|^4_0\\\\N(t)=-(4)^3+6(4)^2+45(2)\\\\N(t)=122$

Hence, $N(t)=-t^3+6t^2+45t$