Question

Diameter of a sphere is increased by 25$\%$. By what percent its C.S.A will increase?

Answer 1

Given that, diameter of a sphere is increased by $25\%$

To find out: Percentage increase in the curved surface area (C.S.A.) of the sphere.

Let the initial diameter of the sphere be $d$.

Hence, increased diameter $=d+25\%\times d$

$d+0.25d=1.25d$

Hence, initial radius $=\frac{d}{2}$

and increased radius $=\frac{1.25d}{2}$

We know that, curved surface area of a sphere $=4\pi r^2$

$\therefore$ Initial C.S.A. $=4\pi {\left(\frac{d}{2}\right)}^2=\pi d^2$

and increased C.S.A. $=4\pi {\left(\frac{1.25d}{2}\right)}^2=\pi (1.25d{)}^2$

$=\left(1.5625\right)\pi d^2$

We know that, $\%\text{increase}=\frac{increased-initial}{initial}\times 100$

$\therefore \%$ increase in C.S.A. $=\frac{1.5625\pi d^2-\pi d^2}{\pi d^2}\times 100$

$=\frac{(1.5625-1)\pi d^2}{\pi d^2}\times 100$

$=56.25\%$

Hence, the required percentage increase in C.S.A. is $56.25\%$.