1. **State the problem:** We need to find the height $h$ of a cylinder given its volume and diameter. 2. **Given:** - Volume $V = 10,000\pi$ in$^3$ - Diameter $d = 32$ in 3. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 4. **Calculate the radius:** $$r = \frac{d}{2} = \frac{32}{2} = 16 \text{ in}$$ 5. **Substitute known values into the volume formula:** $$10,000\pi = \pi \times 16^2 \times h$$ 6. **Simplify the equation:** $$10,000\pi = \pi \times 256 \times h$$ 7. **Divide both sides by $\pi$ to cancel it out:** $$\cancel{\pi} \times 10,000 = \cancel{\pi} \times 256 \times h$$ $$10,000 = 256h$$ 8. **Solve for $h$ by dividing both sides by 256:** $$h = \frac{10,000}{256}$$ 9. **Simplify the fraction:** $$h \approx 39.06$$ 10. **Round to the nearest whole number:** $$h \approx 39 \text{ in}$$ **Final answer:** The height $h$ of the cylinder is approximately 39 inches.