Question

what is the radius of a hemisphere with a volume of 76,386 m³, to the nearest tenth of a meter?

Explanation:

Step1: Recall hemisphere volume formula

The volume $V$ of a hemisphere is $V = \frac{2}{3}\pi r^3$, where $r$ is the radius.

Step2: Substitute given volume

Substitute $V = 76386$ into the formula:
$$76386 = \frac{2}{3}\pi r^3$$

Step3: Isolate $r^3$ term

Multiply both sides by $\frac{3}{2}$:
$$r^3 = \frac{76386 \times 3}{2\pi}$$
Calculate numerator: $76386 \times 3 = 229158$, so
$$r^3 = \frac{229158}{2\pi} = \frac{114579}{\pi}$$

Step4: Calculate $r^3$ value

Use $\pi \approx 3.1416$:
$$r^3 \approx \frac{114579}{3.1416} \approx 36471.5$$

Step5: Solve for $r$

Take the cube root of both sides:
$$r \approx \sqrt[3]{36471.5} \approx 33.2$$

Answer:

$33.2$ meters