Step 1: Understand the formula for the volume of a sphere

The volume of a sphere depends on the cube of its radius: $V = \frac{4}{3}\pi r^3$.

$V = \frac{4}{3}\pi r^3$

Step 2: Determine the new radius after a 50% increase

If the radius is increased by $50\%$, the new radius becomes: $r' = r + 0.50r = 1.5r = \frac{3}{2}r$

$r' = 1.5r$

Step 3: Calculate the new volume

Substitute the new radius into the volume formula: $V' = \frac{4}{3}\pi (1.5r)^3 = \frac{4}{3}\pi (3.375 r^3) = 3.375 V$

$V' = 3.375 V$

Step 4: Find the percentage increase in volume

The percentage increase is given by $\frac{V' - V}{V} \times 100\%$. Since $V' = 3.375V$, the increase is: $(3.375 - 1) \times 100\% = 2.375 \times 100\% = 237.5\%$

\text{Increase percent} = 237.5\