Question: 9.4
Find the volume of the figure. For calculations involving $\pi$, give both the exact value and an approximation to the nearest hundredth.
9.
13. Find the volume of a cube whose side measures 2.5 in.
17. The diameter of the base of a cylinder is 24 cm. The height of the cylinder is 18 cm. Find the volume of the cylinder. Round to the nearest hundredth.
21. The height of a pyramid is 8 m, and the length of a side of the base is 9 m. What is the volume of the pyramid?
25. The diameter of the base of a cylinder is 14 cm. If the volume of the cylinder is 2310cm^3, find the height of the cylinder. Round to the nearest hundredth.
graph: top-center, a sphere-like circle diagram with a horizontal line across the middle labeled 3 cm; the upper portion has a dashed curved arc and the lower portion has a solid curved arc, suggesting a solid formed by two hemispheres or a sphere cross-section.
1. **Problem 9: Volume of a sphere with radius 3 cm**
The figure is a sphere with radius $r=3$ cm.
Formula for volume of a sphere:
$$V=\frac{4}{3}\pi r^3$$
Calculate:
$$V=\frac{4}{3}\pi (3)^3=\frac{4}{3}\pi 27=36\pi$$
Approximate:
$$V\approx 36 \times 3.1416=113.10 \text{ cm}^3$$
2. **Problem 13: Volume of a cube with side 2.5 in**
Formula for volume of a cube:
$$V=s^3$$
Calculate:
$$V=(2.5)^3=15.625 \text{ in}^3$$
3. **Problem 17: Volume of a cylinder with diameter 24 cm and height 18 cm**
Radius $r=\frac{24}{2}=12$ cm
Formula for volume of a cylinder:
$$V=\pi r^2 h$$
Calculate:
$$V=\pi (12)^2 (18)=\pi \times 144 \times 18=2592\pi$$
Approximate:
$$V\approx 2592 \times 3.1416=8131.01 \text{ cm}^3$$
4. **Problem 21: Volume of a pyramid with height 8 m and base side 9 m**
Base is a square with side $9$ m, so area:
$$B=9^2=81 \text{ m}^2$$
Formula for volume of a pyramid:
$$V=\frac{1}{3}Bh$$
Calculate:
$$V=\frac{1}{3} \times 81 \times 8=216 \text{ m}^3$$
5. **Problem 25: Find height of cylinder with diameter 14 cm and volume 2310 cm$^3$**
Radius:
$$r=\frac{14}{2}=7 \text{ cm}$$
Volume formula:
$$V=\pi r^2 h$$
Solve for $h$:
$$h=\frac{V}{\pi r^2}=\frac{2310}{\pi (7)^2}=\frac{2310}{49\pi}$$
Simplify with cancellation:
$$h=\frac{\cancel{2310}}{\cancel{49}\pi} = \frac{2310}{49\pi}$$
Calculate:
$$h=\frac{2310}{49 \times 3.1416} \approx \frac{2310}{153.9384} \approx 15.00 \text{ cm}$$
**Final answers:**
1. Volume of sphere: $36\pi \approx 113.10$ cm$^3$
2. Volume of cube: $15.625$ in$^3$
3. Volume of cylinder: $2592\pi \approx 8131.01$ cm$^3$
4. Volume of pyramid: $216$ m$^3$
5. Height of cylinder: $\approx 15.00$ cm