Answer:
1) Let x be the distance from Alberto to the base of the building. Using trigonometry, we have:
tan(70) = 40 / (x + 6)
x + 6 = 40 / tan(70)
x = 40 / tan(70) - 6
x ≈ 11.1 feet
Therefore, Alberto is about 11.1 feet away from the base of the building.
2) Let h be the height of the building. Using trigonometry, we have:
tan(θ) = h / (100 + 1.55)
h = (100 + 1.55) tan(23)
h ≈ 42.9 feet
Therefore, the height of the building is about 42.9 feet.
3) Let θ be the angle of elevation for each ramp. Using trigonometry, we have:
sin(θ) = 22 / 122
θ = sin⁻¹(22/122)
θ ≈ 10.3 degrees
Therefore, the measure of the angle of elevation for each ramp is about 10.3 degrees.
4) Let d be the distance from the boat to the base of the lighthouse. Using trigonometry, we have:
tan(28) = 40 / d
d = 40 / tan(28)
d ≈ 78.5 feet
Therefore, the distance from the base of the lighthouse to the boat is about 78.5 feet.
5) Let x be the distance from General Han Xin's position to the palace. Using trigonometry, we have:
tan(35) = x / 800
x = 800 tan(35)
x ≈ 595.4 meters
Therefore, the palace is about 595.4 meters away from General Han Xin's position.
6) Let h be the height of the clouds. Using trigonometry, we have:
tan(80) = h / 165
h = 165 tan(80)
h ≈ 702.4 meters
Therefore, the clouds are about 702.4 meters high.
Explanation: