The radius of a sphere is increased by 25%. What is the increase per cent in surface area?

The correct answer is Option 2: 56.25%

Let the original radius of the sphere be $r$.

The formula for the surface area ($A$) of a sphere is given by:
$A=4\pi r^2$

When the radius is increased by 25%, the new radius ($r^{\prime }$) becomes:
$r^{\prime }=r+0.25r=1.25r$

Now, calculate the new surface area ($A^{\prime }$) using the new radius:
$A^{\prime }=4\pi {\left(1.25r\right)}^2$
$A^{\prime }=4\pi \times 1.5625r^2$
$A^{\prime }=1.5625\times \left(4\pi r^2\right)$
$A^{\prime }=1.5625A$

To find the percentage increase in surface area:
$\text{Percentage Increase}=\frac{A^{\prime }-A}{A}\times 100$
$\text{Percentage Increase}=\frac{1.5625A-A}{A}\times 100$
$\text{Percentage Increase}=0.5625\times 100=56.25\%$

Thus, the surface area increases by 56.25%.