The height of the tree is 20.6 feet.
How are the heights and distances problems solved?
To solve problems of heights and distances, we use trigonometric ratios determining the given case by geometric shapes.
How do we solve the given question?
We are given the following information: John who is 5 ft. tall is standing at a distance of 20 ft. from a tree, and you measure the angle of elevation to be 38°. We are asked to find the height of the tree.
To solve the problem, we will use heights and distances.
First, we define a figure:
Let AD be John, such that AD = 5 ft.
Let BE be the tree with its height = h ft.
The distance between John and the tree is determined by DE = 20 ft.
We drop a perpendicular from point A to BE at point C.
Now, ADEC is a rectangle. Hence,
CE = AD = 5 ft.
AC = DE = 20 ft.
We join A and B, to complete the right-angled triangle ACB, with ∠ACB = 90°, ∠BAC = 38° (the given angle of elevation).
In ΔACB,
AC = 20 ft. (already derived)
BC = BE - CE = h-5 ft. (∵ BE = height of the tree h, CE = 5, already derived)
Now, in ΔACB,
tan 38° = BC/AC = (h-5)/20
or, h-5 = 20*tan38° = 20*0.781286 = 15.6257
or, h = 15.6 + 5 = 20.6257.
∴ h = 20.6 (correct to the nearest tenth)
So, the height of the tree is 20.6 feet.
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