Question

The number of square meters in the total surface area of a right circular cylinder, including the top and bottom, is equal to the number of cubic meters in its volume.

If the radius of the cylinder is five times its height, what is its volume?

Answer 1

Find the volume of the cylinder:

Step-1: Find the surface area and volume

Let, $\text{h}$ be the height and $\text{r}$ be the radius of the given cylinder.

If the radius of the cylinder is five times its height, then the radius is $\text{r}=5\text{h}$.

Using the formula of total surface area of cylinder and calculated the total surface in below:

$\begin{array}{rcl}\text{TSA}& =& 2\mathrm{\pi r}\left(\mathrm{h}+\mathrm{r}\right)\left[\text{Total surface area}\mathrm{formula}\right]\\ & \Rightarrow & \mathrm{TSA}=2\mathrm{\pi }\left(5\mathrm{h}\right)\left(\text{h}+5\text{h}\right)\left[\mathrm{Substitute}\mathrm{r}=5\mathrm{h}\right]\\ \therefore \mathrm{TSA}& =& 60{\mathrm{\pi h}}^2...\left(1\right)\left[\mathrm{Simplifying}\right]\end{array}$

Using the formula of volume of cylinder and calculated the volume in below:

$\begin{array}{rcl}\text{V}& =& {\mathrm{\pi r}}^2\mathrm{h}\left[\mathrm{Volume}\mathrm{formula}\right]\\ & \Rightarrow & \mathrm{V}=\mathrm{\pi }{\left(5\mathrm{h}\right)}^2\times \mathrm{h}\left[\mathrm{Substitute}\mathrm{r}=5\mathrm{h}\right]\\ \therefore \mathrm{V}& =& 25{\mathrm{\pi h}}^3...\left(2\right)\left[\mathrm{Simplifying}\right]\end{array}$

Step-2: Calculate the height $\text{h}$ of the given cylinder:

Since the total surface area of the given cylinder is equal to the volume of the given cylinder, so set both equations $\left(1\right)$ and $\left(2\right)$ equal and solve for $\text{h}$:

$\begin{array}{rcl}60{\mathrm{\pi h}}^2& =& 25{\mathrm{\pi h}}^3\left[\mathrm{Equation}\left(1\right)=\mathrm{Equation}\left(2\right)\right]\\ & \Rightarrow & 60=25\mathrm{h}\left[\mathrm{SImplifying}\right]\\ & \Rightarrow & \mathrm{h}=\frac{60}{25}\\ \therefore \mathrm{h}& =& 2.4\end{array}$

Step-3: Calculate the volume of the cylinder.

Substitute $\text{h}=2.4$ in equation $\left(2\right)$ and solve for volume $\text{V}$:

$\begin{array}{rcl}\text{V}& =& 25\mathrm{\pi }{\left(2.4\right)}^3\\ \mathrm{V}& =& 345.6\times 3.14\left[\mathrm{\pi }=3.14\right]\\ \therefore V& =& 1085.184\\ & & \end{array}$

Hence, the volume of the cylinder is $\mathbf{345}\mathbf{.}\mathbf{6}\mathbf{\pi }$ square meters.