To find out how much broth a bowl in the shape of a hemisphere can hold, we need to calculate the volume of the hemisphere. The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.
Given that the diameter of the bowl is 10.2 inches, we can find the radius by dividing the diameter by 2:
\[ r = \frac{10.2}{2} = 5.1 \text{ inches} \]
Now, we can substitute the radius into the volume formula using \( \pi \approx 3.14 \):
\[ V = \frac{2}{3} \times 3.14 \times (5.1)^3 \]
First, we need to calculate \( (5.1)^3 \):
\[ (5.1)^3 = 5.1 \times 5.1 \times 5.1 \] Calculating step by step:
- \( 5.1 \times 5.1 = 26.01 \)
- \( 26.01 \times 5.1 = 132.651 \)
So, \( (5.1)^3 \approx 132.651 \).
Now substitute this back into the volume formula:
\[ V = \frac{2}{3} \times 3.14 \times 132.651 \]
Calculating \( 3.14 \times 132.651 \):
\[ 3.14 \times 132.651 \approx 416.77334 \]
Now compute the full volume:
\[ V = \frac{2}{3} \times 416.77334 \approx 278.51556 \]
Finally, round this to the nearest whole number:
\[ V \approx 279 \]
Thus, the bowl can hold approximately 279 cubic inches of broth.