A sphere of maximum volume is "cut out" from a solid hemisphere of radius r. The ratio of the volume of the hemisphere to that of the cut-out sphere is:

4 : 1

The correct answer is Option (4) → 4 : 1

Step 1: Radius of the sphere

• Let the radius of the hemisphere be r.
• A sphere of maximum volume that can fit inside a hemisphere has radius = r/2.
• This is because the sphere must fit completely inside the hemisphere.

Step 2: Volume of hemisphere

$V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$

Step 3: Volume of sphere

$V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 = \frac{4}{3} \pi \cdot \frac{r^3}{8} = \frac{1}{6}\pi r^3$

Step 4: Ratio

$\frac{V_{\text{hemisphere}}}{V_{\text{sphere}}} = \frac{\frac{2}{3}\pi r^3}{\frac{1}{6}\pi r^3} = \frac{2/3}{1/6} = \frac{2 \cdot 6}{3 \cdot 1} = 4 : 1$