To find the volume of broth that a hemisphere-shaped bowl can hold, we can use the formula for the volume of a hemisphere:
\[ V = \frac{2}{3} \pi r^3 \]
Where \( r \) is the radius of the hemisphere. The radius is half of the diameter.
Given that the diameter of the bowl is 10.2 inches, the radius \( r \) is:
\[ r = \frac{10.2}{2} = 5.1 \text{ inches} \]
Now we can substitute the value of \( r \) into the volume formula:
\[ V = \frac{2}{3} \cdot 3.14 \cdot (5.1)^3 \]
First, we calculate \( (5.1)^3 \):
\[ (5.1)^3 = 5.1 \cdot 5.1 \cdot 5.1 = 132.651 \]
Now substituting back into the volume equation:
\[ V = \frac{2}{3} \cdot 3.14 \cdot 132.651 \]
Calculating \( 3.14 \cdot 132.651 \):
\[ 3.14 \cdot 132.651 \approx 416.05034 \]
Now we multiply this by \( \frac{2}{3} \):
\[ V = \frac{2}{3} \cdot 416.05034 \approx 277.367 \]
Rounding to the nearest whole number, the volume of the bowl is approximately:
\[ \boxed{277} \text{ in}^3 \]
Given the provided options and rounding, the correct answer is 278 in³.