Question

The angle of elevation of the top of a tree is found to be 33° at one point and 59° at a top a point 31 ft. nearer the tree. How high is the tree if both observation points and the base of the tree are in the same horizontal plane?

Answer 1

Condition:

The angle of elevation of the top of a tree is found to be $33{}^{\circ}$ at one point and $59{}^{\circ}$ at a top a point 31 ft. nearer the tree. How high is the tree if both observation points and the base of the tree are in the same horizontal plane?

Solution:

$$S = 33{}^{\circ}, W = 59{}^{\circ}, l = 31 \text{ ft}$$

Triangle DCB $\rightarrow$ tan $S = \frac{h}{l + x} \rightarrow h = (l + x) \tan S$.

Triangle ACB $\rightarrow$ tan $W = \frac{h}{x} \rightarrow h = x \tan W$.

$$x \tan W = (l + x) \tan S.$$ $$x \tan W = l \tan S + x \tan S.$$ $$x \tan W - x \tan S = l \tan S.$$ $$x (\tan W - \tan S) = l \tan S.$$ $$x = \frac{l \tan S}{\tan W - \tan S}.$$ $$h = x \tan W = \frac{l \tan S \tan W}{\tan W - \tan S} = \frac{31 \tan 33 \tan 59}{\tan 59 - \tan 33} = 33.01 \text{ ft} \cong 33 \text{ ft}$$

Answer: 33 ft.