Question

the total time in minutes that the operators at a call center are on the phone with customers $t$ days after the call center goes live is modeled by $f(t)=8t^{3}+15t + 17$. the total number of customers calling $t$ days after the call center goes live is modeled by $c(t)=7t^{2}+9t + 12$. the average time spent on the phone per customer is $g(t)=\frac{f(t)}{c(t)}$. how fast is the average time spent on the phone per customer changing on the ninth day? (round to two decimal places.)

Explanation:

Step1: Recall the quotient - rule

If $g(t)=\frac{f(t)}{c(t)}$, then $g^\prime(t)=\frac{f^\prime(t)c(t)-f(t)c^\prime(t)}{[c(t)]^2}$. First, find $f^\prime(t)$ and $c^\prime(t)$.
$f(t) = 8t^{3}+15t + 17$, so $f^\prime(t)=24t^{2}+15$ using the power - rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$.
$c(t)=7t^{2}+9t + 12$, so $c^\prime(t)=14t + 9$.

Step2: Substitute $f(t), f^\prime(t), c(t), c^\prime(t)$ into the quotient - rule formula

$g^\prime(t)=\frac{(24t^{2}+15)(7t^{2}+9t + 12)-(8t^{3}+15t + 17)(14t + 9)}{(7t^{2}+9t + 12)^{2}}$.

Step3: Expand the numerator

Expand $(24t^{2}+15)(7t^{2}+9t + 12)=24t^{2}(7t^{2}+9t + 12)+15(7t^{2}+9t + 12)$
$=168t^{4}+216t^{3}+288t^{2}+105t^{2}+135t + 180=168t^{4}+216t^{3}+393t^{2}+135t + 180$.
Expand $(8t^{3}+15t + 17)(14t + 9)=8t^{3}(14t + 9)+15t(14t + 9)+17(14t + 9)$
$=112t^{4}+72t^{3}+210t^{2}+135t+238t + 153=112t^{4}+72t^{3}+210t^{2}+373t + 153$.
Then $g^\prime(t)=\frac{168t^{4}+216t^{3}+393t^{2}+135t + 180-(112t^{4}+72t^{3}+210t^{2}+373t + 153)}{(7t^{2}+9t + 12)^{2}}$
$=\frac{168t^{4}+216t^{3}+393t^{2}+135t + 180 - 112t^{4}-72t^{3}-210t^{2}-373t - 153}{(7t^{2}+9t + 12)^{2}}$
$=\frac{56t^{4}+144t^{3}+183t^{2}-238t + 27}{(7t^{2}+9t + 12)^{2}}$.

Step4: Evaluate $g^\prime(t)$ at $t = 9$

First, calculate the denominator:
$c(9)=7\times9^{2}+9\times9 + 12=7\times81+81 + 12=567+81 + 12=660$.
Then calculate the numerator:
$56\times9^{4}+144\times9^{3}+183\times9^{2}-238\times9 + 27$
$=56\times6561+144\times729+183\times81-2142 + 27$
$=367416+104976+14823-2142 + 27$
$=367416+104976+14823+27-2142$
$=472392+14823+27-2142$
$=487215+27-2142$
$=487242-2142=485100$.
$g^\prime(9)=\frac{485100}{660^{2}}=\frac{485100}{435600}\approx1.11$.

Answer:

$1.11$