A sphere has a volume of 2,304π mm³. Find the diameter of the sphere.

Steps to Solve

1. Write the formula for the volume of a sphere

The volume $V$ of a sphere is given by the formula: $$V = \frac{4}{3}\pi r^3$$ where $r$ is the radius of the sphere.

2. Substitute the given volume into the formula

We are given that the volume $V = 2304\pi \text{ mm}^3$. Substitute this into the formula: $$2304\pi = \frac{4}{3}\pi r^3$$

3. Solve for $r^3$

To isolate $r^3$, divide both sides of the equation by $\pi$: $$2304 = \frac{4}{3} r^3$$ Then, multiply both sides by $\frac{3}{4}$: $$2304 \cdot \frac{3}{4} = r^3$$ $$1728 = r^3$$

4. Solve for $r$

Take the cube root of both sides to find $r$: $$r = \sqrt[3]{1728}$$ $$r = 12 \text{ mm}$$

5. Calculate the diameter

The diameter $d$ of the sphere is twice the radius: $$d = 2r = 2(12) = 24 \text{ mm}$$