College Algebra, Chapter 3, 3.7, Section 3.7, Problem 76

Suppose that a tank holds $100$ gallons of water, which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume of water remaining in the tank after $t$ minutes as

$\displaystyle V(t) = 100 \left( 1 - \frac{t}{40} \right)^2$

a.) Find $V^{-1}$. What does $V^{-1}$ represent?

b.) Find $V^{-1} (15)$. What does it represent?



a.) To find $V^{-1}$, we set $y = V(t)$


$
\begin{equation}
\begin{aligned}

y =& 100 \left( 1 - \frac{t}{40} \right)^2
&& \text{Solve for $t$; divide by } 100
\\
\\
\frac{y}{100} =& \left( 1 - \frac{t}{40} \right)^2
&& \text{Take the square root}
\\
\\
1 - \frac{t}{40} =& \pm \sqrt{\frac{y}{100}}
&& \text{Add } \frac{t}{40} \text{ and subtract } \sqrt{\frac{y}{100}}
\\
\\
\frac{t}{40} =& 1 \mp \sqrt{\frac{y}{100}}
&& \text{Multiply by } 40
\\
\\
t =& 40 \left(1 \mp \sqrt{\frac{y}{100}} \right)
&& \text{Interchange $t$ and $y$}
\\
\\
y =& 40 \left( 1 \mp \sqrt{\frac{t}{100}} \right)
&& \text{Simplify}
\\
\\
y =& 40 \left( 1 \mp \frac{\sqrt{t}}{10} \right)
&&

\end{aligned}
\end{equation}
$


Thus, the inverse of $V(t)$ is $\displaystyle V^{-1} (t) = 40 \left( 1 \mp \frac{\sqrt{t}}{10} \right)$.

If $V(t)$ represents the volume of the water remaining in the tank, then $V^{-1} (t)$ represents the volume of the water drained from the tank.

b.) $\displaystyle V^{-1} (15) = 40 \left( 1 \mp \frac{\sqrt{15}}{10} \right)$

$V^{-1} (15) = 40 + 4 \sqrt{15}$ gallons and $V^{-1} (15) = 40 - 4 \sqrt{15}$ gallons $V^{-1} (15)$ represents the volume of the water drained from the tank after 15 minutes. We choose $V^{-1} (15) = 40 + 4 \sqrt{15}$ gallons because if we add the volume of the water remaining and drained from the tank after 15 minutes, the sum is close to $100$ gallons.