Question

a hemisphere fits exactly inside a rectangular prism box with a square base that has edge length 10 inches. what is a reasonable estimate for the volume of the hemisphere?

Explanation:

Step1: Identify hemisphere radius

Since the hemisphere fits exactly in a square-based prism with edge length 10 inches, the diameter of the hemisphere equals the base edge length. Thus, radius $r = \frac{10}{2} = 5$ inches.

Step2: Apply hemisphere volume formula

The volume of a hemisphere is $\frac{2}{3}\pi r^3$. Substitute $r=5$:
$$V = \frac{2}{3}\pi (5)^3$$

Step3: Calculate and estimate value

First compute $5^3 = 125$, then $\frac{2}{3} \times 125 = \frac{250}{3}$. Use $\pi \approx 3.14$:
$$V \approx \frac{250}{3} \times 3.14 \approx 261.67$$

Answer:

A reasonable estimate for the volume of the hemisphere is approximately 262 cubic inches (or ~260 cubic inches if rounding to a simpler whole number estimate).