Cylinder Height 032693

1. **State the problem:** We have a solid made of a cylinder on top of a hemisphere. The total volume is given as $$\frac{2684}{3} \pi$$ cm$^3$. The radius $r$ of both the cylinder and hemisphere is $4$ cm (since diameter is $8$ cm). We need to find the height $h$ of the cylinder. 2. **Formulas:** - Volume of a hemisphere: $$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$ - Volume of a cylinder: $$V_{cylinder} = \pi r^2 h$$ - Total volume: $$V_{total} = V_{cylinder} + V_{hemisphere}$$ 3. **Calculate the volume of the hemisphere:** $$V_{hemisphere} = \frac{2}{3} \pi (4)^3 = \frac{2}{3} \pi \times 64 = \frac{128}{3} \pi$$ 4. **Set up the equation for total volume:** $$\pi r^2 h + \frac{128}{3} \pi = \frac{2684}{3} \pi$$ 5. **Divide both sides by $\pi$ to simplify:** $$r^2 h + \frac{128}{3} = \frac{2684}{3}$$ 6. **Substitute $r=4$:** $$16 h + \frac{128}{3} = \frac{2684}{3}$$ 7. **Subtract $\frac{128}{3}$ from both sides:** $$16 h = \frac{2684}{3} - \frac{128}{3} = \frac{2684 - 128}{3} = \frac{2556}{3}$$ 8. **Simplify the right side:** $$16 h = 852$$ 9. **Divide both sides by 16:** $$h = \frac{852}{16} = \frac{\cancel{852}^{53.25}}{\cancel{16}^{1}} = 53.25$$ 10. **Round to 1 decimal place:** $$h = 53.3 \text{ cm}$$ **Final answer:** The height of the cylinder is $53.3$ cm.