Question

You want to measure the height of a tree growing vertically on the side of a hill.  The hill makes an angle of 20°with the horizontal.  Standing 15ft. downhill from the base of the tree, you measure the angle formed by the hill and the top of the tree to be 45°.  Find the height of the tree to the nearest foot.  Show your work.

Answer 1

By drawing a diagram that represents the problem, obtain triangle ABC such that
m∠A = 45°
AB = 15 ft, distance downhill
Let BC = h, the vertical height of the tree.

Because AB is 20° above the horizontal, m∠C = 90 -(20+45) = 25°.

Apply the Law of Sines to obtain
$\frac{h}{sin45} = \frac{15}{sin25}$
$h=15(\frac{sin45}{sin25} )=25.1$

Answer: 25 ft (nearest foot)