A boy is trying to find the height of a tree. When he stands 95.5 metres away
from the tree (the distance D in Figure 2) the angle shown as è in Figure 2 is
31.5°. The boy’s eyes are 1.15 m from the ground (h in Figure 2). What is the
height of the tree? You can assume that the ground is level and you should
give your answer to an appropriate number of significant figures/decimal
places.
Review your basic trig functions. The height is
1.15 + 95.5tan31.5°
To find the height of the tree, we can use trigonometry. We'll need to use the tangent function.
Let the height of the tree be H.
Given:
Distance from the boy to the tree, D = 95.5 m
Angle, θ = 31.5°
The boy's eye level from the ground, h = 1.15 m
Step 1: Determine the adjacent side of the triangle.
The adjacent side (h1) is the horizontal distance from the tree to the line connecting the boy's eyes to the ground. We can find it by using the cosine function:
cos(θ) = adjacent/hypotenuse
cos(31.5°) = h1/95.5 m
Solving for h1:
h1 = cos(31.5°) * 95.5 m
Step 2: Determine the opposite side of the triangle.
The opposite side (H + h) is the vertical height from the line connecting the boy's eyes to the top of the tree. We can find it by using the tangent function:
tan(θ) = opposite/adjacent
tan(31.5°) = (H + h)/h1
Solving for H + h:
(H + h) = tan(31.5°) * h1
Step 3: Solve for H.
H = (H + h) - h
Substitute the values we have:
H = (tan(31.5°) * h1) - h
Now, let's plug the values and calculate the height of the tree.
Step 4: Calculate the height of the tree.
h1 = cos(31.5°) * 95.5 m
h = 1.15 m
H = (tan(31.5°) * h1) - h
H = (tan(31.5°) * (cos(31.5°) * 95.5 m)) - 1.15 m
Calculating this expression will give you the height of the tree in meters.