Chapter 1.3, Problem 31E

To determine

To find the surface area and volume of the cylinder, if the given cylinder has equal amount of surface area and volume.

Answer to Problem 31E

The surface area and volume of cylinder is $62.5\pi c{m}^{2}or196.35c{m}^2$ .

Explanation of Solution

Given information :

Given cylinder

  

It has equal amount of surface area and volume.

Given cylinder has radius $r=2.5cm$ and height $h=x$

Since surface area is equal to volume

  $\begin{array}{l}\Rightarrow 2\pi rh+2\pi r^2=\pi h{r}^2\\ \Rightarrow 2\pi \left(2.5\right)x+2\pi {\left(2.5\right)}^2=\pi x{\left(2.5\right)}^2\\ \Rightarrow \pi x{\left(2.5\right)}^2-2\pi \left(2.5\right)x=2\pi {\left(2.5\right)}^2\\ \Rightarrow \pi (6.25)x-5\pi x=\left(12.5\right)\pi \\ \Rightarrow 1.25\pi x=\left(12.5\right)\pi \\ \therefore x=10cm\end{array}$

  $\begin{array}{l}\text{Thus, surface area}=2\pi rh+2\pi r^2\\ =2\pi \left(2.5\right)\left(10\right)+2\pi {\left(2.5\right)}^2\\ =50\pi +12.5\pi \\ =62.5\pi c{m}^{2}or196.35c{m}^2\end{array}$

Since surface area and volume are equal.

Hence, the surface area and volume of cylinder is $62.5\pi c{m}^{2}or196.35c{m}^2$ .