Question

How much broth can a bowl in the shape of a hemisphere (half of a sphere) hold? The radius of the bowl is 10 inches. Round your answer to the nearest whole number.

Answer 1

The formula to calculate the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

Since a bowl in the shape of a hemisphere holds half the volume of a sphere, the volume \( V \) of the hemisphere can be calculated using the formula:

\[
V = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3
\]

For this problem, the radius \( r \) of the bowl is given as 10 inches. We can substitute \( r = 10 \) into the hemisphere volume formula:

\[
V = \frac{2}{3} \pi (10)^3
\]

Calculating \( (10)^3 \):

\[
(10)^3 = 1000
\]

Now, substituting this value back into the formula for \( V \):

\[
V = \frac{2}{3} \pi (1000)
\]

\[
V = \frac{2000}{3} \pi
\]

Next, we will compute the numerical value of this expression by approximating \( \pi \approx 3.14159 \):

\[
V \approx \frac{2000}{3} \times 3.14159
\]

Calculating \( \frac{2000}{3} \):

\[
\frac{2000}{3} \approx 666.6667
\]

Now multiplying by \( \pi \):

\[
V \approx 666.6667 \times 3.14159 \approx 2093.883
\]

Now we need to round this result to the nearest whole number:

\[
V \approx 2094
\]

Therefore, the bowl can hold approximately \(\boxed{2094}\) cubic inches of broth.