###### 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*}

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 $$96\pi=16\pi h\text{.}$$

in-context