Question

If the diameter of a sphere is decreased by $25\%$, then its curved surface area decreases by:

• A. $44.25\%$
• B. $75.43\%$
• C. $43.50\%$
• D. $43.75\%$

Answer 1

The correct option is D

$43.75\%$

Let the original diameter of the sphere be $2x$.
Then, original radius of the sphere = $x$
Original curved surface area = $4\pi x^2$
Decreased diameter of the sphere
$=2x-25\%of2x$
$=2x-\frac{x}{2}=\frac{3}{2}x$
Decreased radius of the sphere = $\frac{3}{4}x$
$\therefore$ Decreased curved surface area
$=4\pi {\left(\frac{3}{4}x\right)}^2=\frac{9}{4}\pi x^2$
Decrease in area
$=4\pi x^2-\frac{9}{4}\pi x^2=\frac{7}{4}\pi x^2$
Hence, percentage decrease in area
$=\frac{\frac{7}{4}\pi x^2}{4\pi x^2}\times 100\%$
$=\frac{7}{16}\times 100\%=\frac{175}{4}\%=43.75\%$