Example2.2.11Cylinder Volume

A cylinder's volume is related to its radius and its height by the equation

\begin{equation*} V=\pi r^2h\text{,} \end{equation*}

where \(V\) is the volume, \(r\) is the base's radius, and \(h\) is the height. If we know the volume is 96\(\pi\) cm3 and the radius is 4 cm, then this equation simplifies to

\begin{equation*} 96\pi=16\pi h \end{equation*}
a right cylinder with radius labeled 4 cm; its volume is labeled as 96 pi cm^3
Figure2.2.12A cylinder

Is 4 cm the height of the cylinder? In other words, is \(4\) a solution to \(96\pi=16\pi h\text{?}\) We will substitute \(h\) in the equation with \(4\) to check:

\begin{align*} 96\pi\amp=16\pi h\\ 96\pi\amp\stackrel{?}{=}16\pi \cdot\substitute{4}\\ 96\pi\amp\stackrel{\text{no}}{=}64\pi \end{align*}

Since \(96\pi=64\pi\) is false, \(h=4\) does not satisfy the equation \(96\pi=16\pi h\text{.}\)

Next, we will try \(h=6\text{:}\)

\begin{align*} 96\pi\amp=16\pi h\\ 96\pi\amp\stackrel{?}{=}16\pi \cdot\substitute{6}\\ 96\pi\amp\stackrel{\checkmark}{=}96\pi \end{align*}

When \(h=6\text{,}\) the equation \(96\pi=16\pi h\) is true. This tells us that \(6\) is a solution for \(h\) in the equation \(96\pi=16\pi h\text{.}\)