Define Initial Surface Area

Let the original radius of the sphere be $r$. The initial surface area $S_1$ of the sphere is given by the formula $S_1 = 4\pi r^2$.

$S_1 = 4\pi r^2$

Calculate New Radius

When the radius is increased by $50\%$, the new radius $r'$ becomes: $r' = r + 0.50r = 1.5r$

$r' = 1.5r$

Calculate New Surface Area

Substitute the new radius $r'$ into the surface area formula to find the new surface area $S_2$: $S_2 = 4\pi (r')^2 = 4\pi (1.5r)^2 = 4\pi (2.25r^2) = 2.25 \times (4\pi r^2) = 2.25 S_1$

$S_2 = 2.25 S_1$

Determine Percent Increase

The percent increase in surface area is calculated as: $\text{Percent Increase} = \frac{S_2 - S_1}{S_1} \times 100\% = \frac{2.25 S_1 - S_1}{S_1} \times 100\% = 1.25 \times 100\% = 125\%$

$\text{Percent Increase} = 125\%$