Question

pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. the angle of elevation of the tree from one position on a flat path from the tree is 50°, and from a second position 50 feet farther along this path it is 20°. what is the height of the tree? the height of the tree is approximately ft.

Explanation:

Step1: Let the height of the tree be $h$ feet and the distance from the first - position to the base of the tree be $x$ feet.

We know that from the first - position, $\tan(50^{\circ})=\frac{h}{x}$, so $h = x\tan(50^{\circ})$. From the second - position, the distance from the tree is $(x + 50)$ feet, and $\tan(20^{\circ})=\frac{h}{x + 50}$, so $h=(x + 50)\tan(20^{\circ})$.

Step2: Since $x\tan(50^{\circ})=(x + 50)\tan(20^{\circ})$.

Expand the right - hand side: $x\tan(50^{\circ})=x\tan(20^{\circ})+50\tan(20^{\circ})$.
Rearrange the terms to solve for $x$: $x\tan(50^{\circ})-x\tan(20^{\circ})=50\tan(20^{\circ})$.
Factor out $x$: $x(\tan(50^{\circ})-\tan(20^{\circ}))=50\tan(20^{\circ})$.
Then $x=\frac{50\tan(20^{\circ})}{\tan(50^{\circ})-\tan(20^{\circ})}$.
We know that $\tan(50^{\circ})\approx1.1918$ and $\tan(20^{\circ})\approx0.3640$.
$x=\frac{50\times0.3640}{1.1918 - 0.3640}=\frac{18.2}{0.8278}\approx22$.

Step3: Calculate the height of the tree $h$.

Since $h = x\tan(50^{\circ})$, substituting $x\approx22$ and $\tan(50^{\circ})\approx1.1918$, we get $h\approx22\times1.1918\approx26$.

Answer:

26