Question

1. a technician can test video player chips at the rate of -3t² + 16t + 5 chips per hour (for 0 ≤ t ≤ 6), where t is the number of hours after 9:00 am. how many chips can be tested between 11:00 a.m. to 3:00 p.m.? (assume t = 0 for 9:00 a.m.) (show work)

Explanation:

Step1: Determine the limits of integration

At 11:00 a.m., $t = 2$ (since $t = 0$ at 9:00 a.m.). At 3:00 p.m., $t=6$.

Step2: Set up the definite - integral

The number of chips tested $N$ is given by the definite integral of the rate function $r(t)=-3t^{2}+16t + 5$ from $t = 2$ to $t = 6$. So $N=\int_{2}^{6}(-3t^{2}+16t + 5)dt$.

Step3: Find the antiderivative

The antiderivative of $-3t^{2}$ is $-t^{3}$, the antiderivative of $16t$ is $8t^{2}$, and the antiderivative of $5$ is $5t$. So the antiderivative of $-3t^{2}+16t + 5$ is $F(t)=-t^{3}+8t^{2}+5t$.

Step4: Evaluate the definite - integral

Using the fundamental theorem of calculus $\int_{a}^{b}f(t)dt=F(b)-F(a)$, we have $F(6)-F(2)$.
$F(6)=-(6)^{3}+8\times(6)^{2}+5\times6=-216 + 288+30=102$.
$F(2)=-(2)^{3}+8\times(2)^{2}+5\times2=-8 + 32 + 10=34$.
$F(6)-F(2)=102 - 34=68$.

Answer:

68