1. **State the problem:** We need to find the radius $r$ of a hemisphere given its volume $V = 78100$ cm$^3$. 2. **Formula for volume of a hemisphere:** The volume of a full sphere is given by $$V_{sphere} = \frac{4}{3} \pi r^3.$$ Since a hemisphere is half of a sphere, its volume is $$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3.$$ 3. **Set up the equation:** Given $V_{hemisphere} = 78100$, we have $$78100 = \frac{2}{3} \pi r^3.$$ 4. **Solve for $r^3$:** Multiply both sides by $\frac{3}{2}$ to isolate $\pi r^3$: $$\cancel{\frac{3}{2}} \times 78100 = \cancel{\frac{3}{2}} \times \frac{2}{3} \pi r^3 \implies \frac{3}{2} \times 78100 = \pi r^3.$$ Calculate the left side: $$\frac{3}{2} \times 78100 = 1.5 \times 78100 = 117150.$$ So, $$117150 = \pi r^3.$$ 5. **Divide both sides by $\pi$ to solve for $r^3$:** $$\frac{117150}{\pi} = r^3.$$ 6. **Calculate $r^3$ numerically:** $$r^3 = \frac{117150}{3.1416} \approx 37279.5.$$ 7. **Find $r$ by taking the cube root:** $$r = \sqrt[3]{37279.5} \approx 33.3 \text{ cm}.$$ **Final answer:** The radius of the hemisphere is approximately **33.3 cm** (to 1 decimal place).