1. The height of the hill is 460 feet.
- We are given the angle of elevation (49°) and the adjacent side (400 ft).
- Use the tangent function: tan(θ) = opposite / adjacent.
- tan(49°) = height / 400.
- height = 400 * tan(49°).
- height ≈ 400 * 1.1504 ≈ 460.16.
- Rounded to the nearest whole number: 460 ft.
2. The angle of elevation of the sun is 35 degrees.
- We are given the opposite side (12.5 m) and the adjacent side (18 m).
- Use the tangent function: tan(θ) = opposite / adjacent.
- tan(θ) = 12.5 / 18 ≈ 0.6944.
- θ = arctan(0.6944) ≈ 34.78°.
- Rounded to the nearest whole degree: 35°.
3. The ladder is 24 feet long.
- We are given the angle (78°) and the adjacent side (5 ft).
- Use the cosine function: cos(θ) = adjacent / hypotenuse.
- cos(78°) = 5 / length.
- length = 5 / cos(78°).
- length ≈ 5 / 0.2079 ≈ 24.05.
- Rounded to the nearest whole number: 24 ft.
4. The angle of elevation is 68 degrees.
- The vertical distance from the person's eyes to the controller is 132 - 5 = 127 ft.
- The horizontal distance is 100 ft.
- Use the tangent function: tan(θ) = opposite / adjacent.
- tan(θ) = 127 / 100 = 1.27.
- θ = arctan(1.27) ≈ 51.78°.
- Wait, let me recalculate: 127/100 = 1.27, arctan(1.27) is approximately 51.8°, which rounds to 52°.
- Correction: Let me recheck the calculation.
- Actually, arctan(1.27) is approximately 51.8°, so rounded to the nearest degree it is 52°.
- But let me verify with a calculator: tan(52°) ≈ 1.2799, which is very close to 1.27.
- So the correct answer should be 52°.
- However, let me double-check the problem setup. The height difference is 132 - 5 = 127 ft, horizontal is 100 ft, so tan(θ) = 127/100 = 1.27, θ = arctan(1.27) ≈ 51.78°, which rounds to 52°.
- Therefore, the angle of elevation is 52 degrees.
Parent Tip: Review the logic above to help your child master the concept of worksheet angles of depression and elevation.
