1. **Problem 6: Combined effect of curvature and refraction on level sights** Given distances: 30, 50, 100, 250, 300, 500, 1000, 2000 m The combined effect $h$ in meters is approximately given by the formula: $$h = 0.0675 d^2$$ where $d$ is the distance in kilometers (km). Convert meters to kilometers: $d = \frac{distance}{1000}$ Calculate $h$ for each distance: - For 30 m: $d=0.03$ km, $h=0.0675 \times (0.03)^2=0.00006075$ m - For 50 m: $d=0.05$ km, $h=0.0675 \times (0.05)^2=0.00016875$ m - For 100 m: $d=0.1$ km, $h=0.0675 \times (0.1)^2=0.000675$ m - For 250 m: $d=0.25$ km, $h=0.0675 \times (0.25)^2=0.00421875$ m - For 300 m: $d=0.3$ km, $h=0.0675 \times (0.3)^2=0.006075$ m - For 500 m: $d=0.5$ km, $h=0.0675 \times (0.5)^2=0.016875$ m - For 1000 m: $d=1$ km, $h=0.0675 \times (1)^2=0.0675$ m - For 2000 m: $d=2$ km, $h=0.0675 \times (2)^2=0.27$ m 2. **Problem 7: Difference in elevation with curvature and refraction** Given: - Backsight (BS) = 3.055 m at 75 m from station - Foresight (FS) = 1.258 m at same station Calculate combined correction of curvature and refraction: Distance $d=75$ m = 0.075 km $$h = 0.0675 \times (0.075)^2 = 0.000380625\text{ m}$$ Difference in elevation ignoring curvature: $$\Delta E = BS - FS = 3.055 - 1.258 = 1.797\text{ m}$$ Adjusted difference in elevation: $$\Delta E_{adj} = \Delta E + h = 1.797 + 0.000380625 = 1.7974\text{ m (approx)}$$ 3. **Problem 8: Backsight or foresight distance for given errors due to curvature and refraction** Formula: $$h = 0.0675 d^2 \Rightarrow d = \sqrt{\frac{h}{0.0675}}$$ Calculate $d$ for each $h$: - For $h=0.0015$ m: $d=\sqrt{\frac{0.0015}{0.0675}}=0.149$ km = 149 m - $h=0.0575$: $d=\sqrt{\frac{0.0575}{0.0675}}=0.918$ km = 918 m - $h=0.0986$: $d=\sqrt{\frac{0.0986}{0.0675}}=1.208$ km = 1208 m - $h=0.2935$: $d=\sqrt{\frac{0.2935}{0.0675}}=2.086$ km = 2086 m - $h=0.8750$: $d=\sqrt{\frac{0.875}{0.0675}}=3.606$ km = 3606 m 4. **Problem 9: Height of lighthouse above sea level** Given: - Distance $d = 16.5$ km - Eye height of observer $h_o = 1.735$ m Formula for distance to horizon: $$d = \sqrt{13 h}$$ where $d$ in km and $h$ in m. Calculate horizon distance of observer: $$d_o = \sqrt{13 \times 1.735} = \sqrt{22.555} = 4.75\text{ km}$$ Height of lighthouse corresponds to horizon distance: $$d_l = d - d_o = 16.5 - 4.75 = 11.75\text{ km}$$ Height of lighthouse: $$h_l = \frac{d_l^2}{13} = \frac{(11.75)^2}{13} = \frac{138.06}{13} = 10.62\text{ m}$$ Total height of lighthouse above sea level: $$H = h_l + h_o = 10.62 + 1.735 = 12.355\text{ m}$$ 5. **Problem 10: Distance out from shore for vessel light to disappear** Given: - Height of light on vessel $h_v=9.45$ m - Eye height of child $h_c=1.32$ m Calculate distances to horizon for both: $$d_v = \sqrt{13 \times 9.45} = \sqrt{122.85} = 11.08 \text{ km}$$ $$d_c = \sqrt{13 \times 1.32} = \sqrt{17.16} = 4.14 \text{ km}$$ Total distance when light disappears: $$D = d_v + d_c = 11.08 + 4.14 = 15.22 \text{ km}$$ 6. **Problem 11: Height of shortest visible tree across lake** Given: - Width of lake $d=24$ km - Eye height of observer $h_o=1.675$ m Observer horizon distance: $$d_o = \sqrt{13 \times 1.675} = \sqrt{21.775} = 4.67 \text{ km}$$ Distance tree is from observer: $$d_t = d - d_o = 24 - 4.67 = 19.33 \text{ km}$$ Height of tree tip: $$h_t = \frac{d_t^2}{13} = \frac{(19.33)^2}{13} = \frac{373.50}{13} = 28.73 \text{ m}$$ Total height above shore: $$H = h_t + h_o = 28.73 + 1.675 = 30.405 \text{ m}$$ 7. **Problem 12: Distance between two towers with equal base elevations** Given: - Eye height of person on tower A: $h=15.5$ m Distance to horizon from tower A: $$d = \sqrt{13 \times 15.5} = \sqrt{201.5} = 14.19 \text{ km}$$ Since bases are level and person on tower A just sees top of B, distance between towers: $$D = 2d = 2 \times 14.19 = 28.38 \text{ km}$$ 8. **Problem 13: Distance a life raft can go before disappearing from lifeguard's sight** Given: - Eye height of lifeguard $h_l=3.5$ m - Life raft at water level Distance to horizon for lifeguard: $$D = \sqrt{13 \times 3.5} = \sqrt{45.5} = 6.75 \text{ km}$$ Thus, the raft disappears after approximately 6.75 km from shore. --- **Final answers:** - Problem 6: Effects (m) = [0.00006, 0.00017, 0.00068, 0.00422, 0.00608, 0.01688, 0.0675, 0.27] - Problem 7: Difference in elevation approx 1.7974 m - Problem 8: Distances (m) = [149, 918, 1208, 2086, 3606] - Problem 9: Lighthouse height approx 12.36 m - Problem 10: Distance light disappears = 15.22 km - Problem 11: Tree height approx 30.41 m - Problem 12: Distance between towers approx 28.38 km - Problem 13: Raft disappearance distance approx 6.75 km