Q. 72 - Precalculus Enhanced with Graphing Utilities

Question

The volume $V$of a right circular cone is $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}.$If the height is twice the radius, express the volume $V$ as a function of $r$. 

Step-by-Step Solution

Answer

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

Step 1. Given Information

Given that $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ and $h\left(r\right)=2r.$

Step 2. Solution

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

Here, $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}, \mathrm{h}\left(\mathrm{r}\right)=2\mathrm{r}.$

Now,

$$\begin{gathered} V\circ h\left(r\right)=V\left(h\right(r\left)\right) \\ V\circ h\left(r\right)=\frac{1}{3}{\mathrm{\pi r}}^2\left\{\mathrm{h}\right(\mathrm{r}\left)\right\} \\ \mathrm{V}\circ \mathrm{h}\left(\mathrm{r}\right)=\frac{1}{3}{\mathrm{\pi r}}^2\left\{2\mathrm{r}\right\} \\ \mathrm{V}\circ \mathrm{h}\left(\mathrm{r}\right)=\frac{2}{3}{\mathrm{\pi r}}^3 \\ \mathrm{V}\left(\mathrm{r}\right)=\frac{2}{3}{\mathrm{\pi r}}^3 \end{gathered}$$

Step 3. Final answer

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