1. **Problem statement:** We have a shape made from a cylinder and a hemisphere. The radius $r$ of both is 8 cm. The total volume is given as $\frac{2684}{3} \pi$ cm$^3$. We need to find the height $h$ of the cylinder. 2. **Formulas:** - Volume of a cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of a sphere: $$V_{sphere} = \frac{4}{3} \pi r^3$$ - Volume of a hemisphere (half a sphere): $$V_{hemi} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$ 3. **Given values:** - Radius $r = 8$ cm - Total volume $V_{total} = \frac{2684}{3} \pi$ 4. **Calculate the volume of the hemisphere:** $$V_{hemi} = \frac{2}{3} \pi (8)^3 = \frac{2}{3} \pi \times 512 = \frac{1024}{3} \pi$$ 5. **Express total volume as sum of cylinder and hemisphere volumes:** $$V_{total} = V_{cyl} + V_{hemi}$$ $$\frac{2684}{3} \pi = \pi r^2 h + \frac{1024}{3} \pi$$ 6. **Divide both sides by $\pi$ to simplify:** $$\frac{2684}{3} = 64 h + \frac{1024}{3}$$ 7. **Isolate $h$:** $$64 h = \frac{2684}{3} - \frac{1024}{3} = \frac{2684 - 1024}{3} = \frac{1660}{3}$$ $$h = \frac{1660}{3 \times 64} = \frac{1660}{192}$$ 8. **Calculate $h$ numerically:** $$h \approx 8.6458$$ 9. **Round to 1 decimal place:** $$h \approx 8.6 \text{ cm}$$ **Final answer:** The height of the cylinder is approximately 8.6 cm.