1. **State the problem:** We need to find the volume of a sphere with radius $r = 5$ inches. 2. **Formula for the volume of a sphere:** The volume $V$ of a sphere is given by the formula: $$V = \frac{4}{3} \pi r^3$$ 3. **Substitute the radius:** Plug in $r = 5$ inches into the formula: $$V = \frac{4}{3} \pi (5)^3$$ 4. **Calculate the cube of the radius:** $$5^3 = 5 \times 5 \times 5 = 125$$ 5. **Rewrite the volume expression:** $$V = \frac{4}{3} \pi \times 125$$ 6. **Multiply numerator terms:** $$V = \frac{4 \times 125}{3} \pi = \frac{500}{3} \pi$$ 7. **Simplify the fraction:** $$V = \frac{\cancel{500}}{\cancel{3}} \pi$$ (Note: 500 and 3 have no common factors, so fraction remains $\frac{500}{3}$.) 8. **Calculate the numerical value:** $$V \approx \frac{500}{3} \times 3.1416 \approx 523.598$$ 9. **Round to the nearest hundredth:** $$V \approx 523.60$$ cubic inches. **Final answer:** The volume of the sphere is approximately **523.60 cubic inches**.