Question

The radius of a hemispherical balloon increases from $6cm$ to $12cm$ as air is being pumped into it$.$ The ratio of the surface areas of the balloon in the two cases are

• A. $1:2$
• B. $4:2$
• C. $2:3$
• D. $1:4$

Answer 1

The correct option is D

$1:4$

It is given that$,$

The radius of a hemispherical balloon increases from $6cm$ to $12cm$

Let $r_1=6cm$

and $r_2=12cm$

We know that total surface area of hemisphere $=3{\mathrm{\pi r}}^2$

Ratio of the surface areas of the balloon in the two cases $=\frac{3{{\mathrm{\pi r}}_1}^2}{3{{\mathrm{\pi r}}_2}^2}$

$={\left(\frac{r_1}{r_2}\right)}^2$

$={\left(\frac{6}{12}\right)}^2$

$={\left(\frac{1}{2}\right)}^2$

$=\frac{1}{4}$

Therefore, the ratio of the surface areas of the balloon in the two cases are $1:4$

Hence, option $\left(D\right)$ is correct.