1. Problem 22 states: Given the Earth's mean radius $r_\text{Earth} = 6.37 \times 10^{6} \ m$ and Moon's mean radius $r_\text{Moon} = 1.74 \times 10^{8} \ cm$, calculate: (a) The ratio of Earth's surface area to Moon's surface area. (b) The ratio of Earth's volume to Moon's volume. Note: Convert Moon's radius to meters because SI units must be consistent. 2. Convert Moon's radius from cm to m: $$r_\text{Moon} = 1.74 \times 10^{8} \ {cm} \times \left( \frac{1 \ m}{100 \ cm} \right) = 1.74 \times 10^{6} \ m$$ 3. Recall surface area formula for a sphere: $$S = 4 \pi r^{2}$$ Calculate the surface areas: $$S_\text{Earth} = 4 \pi (6.37 \times 10^{6})^{2}$$ $$S_\text{Moon} = 4 \pi (1.74 \times 10^{6})^{2}$$ Find the ratio: $$\frac{S_\text{Earth}}{S_\text{Moon}} = \frac{4 \pi (6.37 \times 10^{6})^{2}}{4 \pi (1.74 \times 10^{6})^{2}} = \left( \frac{6.37 \times 10^{6}}{1.74 \times 10^{6}} \right)^{2} = \left( \frac{6.37}{1.74} \right)^{2}$$ Calculate numeric value: $$\frac{6.37}{1.74} \approx 3.66$$ So, $$\frac{S_\text{Earth}}{S_\text{Moon}} = 3.66^{2} = 13.4$$ 4. Recall volume formula for a sphere: $$V = \frac{4}{3} \pi r^{3}$$ Calculate volumes: $$V_\text{Earth} = \frac{4}{3} \pi (6.37 \times 10^{6})^{3}$$ $$V_\text{Moon} = \frac{4}{3} \pi (1.74 \times 10^{6})^{3}$$ Find the ratio: $$\frac{V_\text{Earth}}{V_\text{Moon}} = \frac{(6.37 \times 10^{6})^{3}}{(1.74 \times 10^{6})^{3}} = \left( \frac{6.37}{1.74} \right)^{3} = 3.66^{3}$$ Calculate numeric value: $$3.66^{3} = 49.0$$ 5. Final answers: (a) Ratio of surface areas = $13.4$ (b) Ratio of volumes = $49.0$ These ratios show Earth has about 13.4 times the surface area and 49 times the volume of the Moon.