Question

The total surface area of a solid cylinder is $616{\text{cm}}^2$. If the ratio between its curved surface area and total surface area is $1:2,$ find the volume of the cylinder.

Answer 1

Let $S_1=$ curved surface area

Radius $=r$

$S_2=$ total surface area

Height $=h$

$\begin{array}{c}\frac{S_1}{S_2}=\frac{1}{2}\\ \Rightarrow S_1=\frac{1}{2}\times S_2\\ \Rightarrow S_1=\frac{1}{2}\times 616=308{\text{cm}}^2\\ \text{again},\frac{S_1}{S_2}=\frac{1}{2}\\ \frac{2\pi rh}{2\pi rh+2\pi r^2}=\frac{1}{2}\\ \frac{2\pi rh}{2\pi r\left[h+r\right]}=\frac{1}{2}\\ \Rightarrow \frac{h}{h+r}=\frac{1}{2}\\ \Rightarrow 2h=h+r\\ \Rightarrow 2h-h=r\\ \Rightarrow h=r\\ S_1=2\pi rh=2\times \frac{22}{7}\times r\times r=308{\text{cm}}^2\\ r^2=\frac{308\times 7}{2\times 22}{\text{cm}}^2\\ r^2=49{\text{cm}}^2\\ r=7\text{cm}\\ r=h\\ h=7\text{cm}\\ Volume=\pi r^{2}h\\ =\frac{22}{7}\times 7\times 7\times 7\\ =1078{\text{cm}}^3\end{array}$