Question

A cone hemisphere and a cylinder stands on equal bases and have the same height the height being equal to the radius of the circular base then their whole surface areas are in the ratio

• A. $(\sqrt{2}+1):3:4$
• B. $(\sqrt{3}+1):3:4$
• C. $\sqrt{2}:3:4$
• D. $\sqrt{3}:7:8$

Answer 1

Answer: (A)

$(\sqrt{2}+1):3:4$

Let $r$ and $h$ be the radius of base and height of the cylinder, cone and hemisphere.

We know that, Height of hemisphere $=$ Radius of the hemisphere

$\therefore h=r$

Whole surface area of cylinder $=2\pi r(r+h)=2\pi r(r+r)=4\pi r^2$

Whole surface area of cone $=\pi r(r+\sqrt{r^2+h^2})=\pi r(r+\sqrt{r^2+r^2})=\pi r(r+\sqrt{2}r)=\pi r^2(1+\sqrt{2})$

Whole surface area of hemisphere $=3\pi r^2$

$\therefore$ Whole surface area of cone $:$ Whole surface area of hemisphere $:$ Whole surface area of cylinder

$=\pi r^2(1+\sqrt{2}):3\pi r^2:4\pi r^2$

$=(\sqrt{2}+1):3:4$

So, $\text{A}$ is the correct option.