Question

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

• A. $63.75\%$
• B. $43.95\%$
• C. $73.75\%$
• D. $43.75\%$

Answer 1

The correct option is D

$43.75\%$

Let the diameter of the sphere be d.
Radius $\left(r_1\right)$ of the sphere $=\frac{d}{2}$
New radius, $\left(r_2\right)$ of the sphere $=\frac{d}{2}(1-\frac{25}{100})=\frac{3}{8}d$
CSA $\left(S_1\right)$ of sphere $=4\pi r_1^2$
$=4\pi {\left(\frac{d}{2}\right)}^2=\pi d^2$
CSA $\left(S_2\right)$ of the sphere when radius is decreased $=4\pi r_2^2$
$=4\pi {\left(\frac{3d}{8}\right)}^2=\frac{9}{16}\pi d^2$
Decrease in the surface area of the sphere $=S_1-S_2$
$=\pi d^2-\frac{9}{16}\pi d^2=\frac{7}{16}\pi d^2$
Percentage decrease in the surface area of the sphere $=\frac{S_1-S_2}{S_1}\times 100$
$=\frac{7\pi d^2}{16\pi d^2}\times 100=\frac{700}{16}=43.75\%$