Q. 71 - Precalculus Enhanced with Graphing Utilities

Question

The volume $V$ of a right circular cylinder of height $h$ and radius $r$ is $V={\mathrm{\pi r}}^2\mathrm{h}.$If the height is twice the radius, express the volume $V$ as a function of $r$. 

Step-by-Step Solution

Answer

The volume, $V\left(r\right)=2{\mathrm{\pi r}}^3.$

Step 1. Given Information

Given that $V={\mathrm{\pi r}}^2\mathrm{h}$ and $h\left(r\right)=2r.$

Step 2. Solution

We have to find $V$ as a function of $r$.

Here, $V={\mathrm{\pi r}}^2\mathrm{h}; \mathrm{h}\left(\mathrm{r}\right)=2\mathrm{r}.$

Now,

$$\begin{gathered} V\circ h\left(r\right)=V\left(h\right(r\left)\right) \\ V\circ h\left(r\right)={\mathrm{\pi r}}^2\left\{\mathrm{h}\right(\mathrm{r}\left)\right\} \\ \mathrm{V}\circ \mathrm{h}\left(\mathrm{r}\right)={\mathrm{\pi r}}^2\left(2\mathrm{r}\right) \\ \mathrm{V}\circ \mathrm{h}\left(\mathrm{r}\right)=2{\mathrm{\pi r}}^3 \\ \mathrm{V}\left(\mathrm{r}\right)=2{\mathrm{\pi r}}^3. \end{gathered}$$

Step 3. Final Answer

Hence, $V\left(r\right)=2{\mathrm{\pi r}}^3.$