Question

From a solid sphere of radius $R$, a concentric solid sphere of radius $\frac{R}{2}$ is removed. The total surface area increases by

• A. $0\%$
• B. $25\%$
• C. $50\%$
• D. $10\%$

Answer 1

The correct option is B $25\%$

Given,

From a solid sphere of radius $R$, a concentric solid sphere of radius $\frac{R}{2}$ is removed.

Initial surface area of the sphere $\left(A_1\right)=4\pi R^2$

Now, the new surface area of sphere $\left(A_2\right)=4\pi R^2+4\pi {\left(\frac{R}{2}\right)}^2$

$=4\pi R^2+\pi R^2$

$=5\pi R^2$.

$\therefore$ Increase in area is $=\frac{A_2-A_1}{A_1}\times 100\%$

$=\frac{5\pi R^2-4\pi R^2}{4\pi R^2}\times 100\%$

$=\frac{\pi R^2}{4\pi R^2}\times 100\%$

$=25\%$.

Hence, the area will be increased by $25\%$.

Therefore, option $B$ is correct.