Answer
$a)$ For $75$ feet the maximum time is $39.18$ minutes and for $95$ feet is $23.04$ minutes.
$b)$ The maximum depth possible in a dive of $20$ minutes is $100.32$ feet.
Work Step by Step
$a)$ In this exercise we just need to evaluate the value of the function when $x=75$ and $x=95$:
$t(75)=286.93(0.9738)^{75}\approx39.18 $ minutes
$t(95)=286.93(0.9738)^{95}\approx23.04 $ minutes
$b)$ Here we just need to solve the following equation $t(x)=20$, so then we have:
$t(x)=20 ⇔ 286.93(0.9738)^{x}=20⇔(0.9738)^{x}=\frac{20}{286.93}⇔$
$⇔e^{x\ln0.9738}=\frac{20}{286.93}⇔x\ln0.9738=\ln(\frac{20}{286.93})⇔$
$⇔x=\ln(\frac{20}{286.93}):\ln0.9738⇔x\approx100.32 $ feet
