Final answer:
Using trigonometric ratios and simultaneous equations, the height of the tree is found to be approximately 14.56 meters when given two angles of elevation (19 degrees and 32 degrees) and the horizontal distance between two observation points (18 meters apart).
Step-by-step explanation:
To work out the height of the tree using the angles of elevation from points A and B, we need to apply trigonometry using the tangent ratios of the angles given. Let's assume the height of the tree is h meters, and the distance from point A to the base of the tree is x meters. Then, the distance from point B to the base of the tree is x - 18 meters, since point B is 18m closer to the tree than point A.
From point A, with an angle of elevation of 19 degrees, the tangent ratio gives us:
tan(19°) = h / x
From point B, with an angle of elevation of 32 degrees, the tangent ratio gives us:
tan(32°) = h / (x - 18)
We now have two equations with two variables, which can be solved simultaneously:
- h = x * tan(19°)
- h = (x - 18) * tan(32°)
Setting the two expressions for h equal to each other and solving for x, we get:
x * tan(19°) = (x - 18) * tan(32°)
After the calculations, we find that x approximately equals 42.96 meters and h approximately equals 14.56 meters. Therefore, the correct answer is d) 14.56 meters.