Questions: A punch bowl is in the shape of a hemisphere with a radius of 10 inches. The cup part of a ladle is also in the shape of a hemisphere with a radius of 4 inches. If the bowl is full, how many full ladles of punch are there in the bowl? O 15 O 16 O 2 O None of these O 3

Solution Steps

The punch bowl is a hemisphere with a radius of 10 inches. The volume \( V \) of a hemisphere is given by:

\[ V = \frac{2}{3} \pi r^3 \]

Substituting \( r = 10 \):

\[ V_{\text{bowl}} = \frac{2}{3} \pi (10)^3 = \frac{2}{3} \pi \times 1000 = \frac{2000}{3} \pi \]

The ladle is also a hemisphere with a radius of 4 inches. Using the same formula for the volume of a hemisphere:

\[ V_{\text{ladle}} = \frac{2}{3} \pi (4)^3 = \frac{2}{3} \pi \times 64 = \frac{128}{3} \pi \]

To find the number of full ladles that can be filled from the punch bowl, divide the volume of the bowl by the volume of the ladle:

\[ \text{Number of ladles} = \frac{V_{\text{bowl}}}{V_{\text{ladle}}} = \frac{\frac{2000}{3} \pi}{\frac{128}{3} \pi} = \frac{2000}{128} = \frac{125}{8} = 15.625 \]

Since only full ladles are counted, we take the integer part of the result:

\[ \text{Number of full ladles} = 15 \]

Final Answer