Question

The diameter of a sphere is decreased by $25$%. By what percent does its curved surface area decrease?

• A. $53.85\%$
• B. $34.85\%$
• C. $45.65\%$
• D. $43.75\%$

Answer 1

The correct option is D $43.75\%$

Let diameter and radius of sphere be $d$ and $r$, respectively.

Then, curved surface area of sphere $4\pi r^2=4\pi {\left(\frac{d}{2}\right)}^2=\pi d^2$.

Diameter decreased by $25\%=$ $\frac{25}{100}=\frac{1}{4}$.

$\therefore$ New diameter $=d-\frac{d}{4}=\frac{3d}{4}$.

Then, the new curved surface area $=\pi {\left(\frac{3d}{4}\right)}^2=\frac{9}{16}\pi d^2$.

$\therefore \%$ decrease in the curved surface area of the sphere $=\frac{\pi d^2-\frac{9}{16}\pi d^2}{\pi d^2}\times \frac{100}{100}=\frac{700}{16\times 100}=43.75\%$.

Hence, option $D$ is correct.