1. **State the problem:** We need to find the volume of a sphere with radius $r = 15$ meters. 2. **Formula for the volume of a sphere:** $$V = \frac{4}{3} \pi r^3$$ This formula calculates the volume by multiplying four-thirds by pi and the cube of the radius. 3. **Substitute the radius into the formula:** $$V = \frac{4}{3} \pi (15)^3$$ 4. **Calculate the cube of the radius:** $$15^3 = 15 \times 15 \times 15 = 3375$$ 5. **Plug this value back into the volume formula:** $$V = \frac{4}{3} \pi \times 3375$$ 6. **Multiply the constants:** $$V = \frac{4}{3} \times 3375 \times \pi$$ 7. **Simplify the fraction:** $$\frac{4}{3} \times 3375 = \cancel{\frac{4}{3}} \times 3375 = 4 \times 1125 = 4500$$ 8. **So the volume is:** $$V = 4500 \pi$$ 9. **Calculate the numerical value using $\pi \approx 3.1416$:** $$V \approx 4500 \times 3.1416 = 14137.17$$ 10. **Round to the nearest hundredth:** $$\boxed{14137.17 \text{ cubic meters}}$$