the total time in minutes that the operators at a call center are on the phone with customers $t$ days after…

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.)

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.)

Answer

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:

$1.11$