Question

as shown in the figure below, tom is looking at a tree that is leaning at an angle of 78° with respect to the ground. the angle of elevation from where tom is standing to the top of the tree is 30°. the length of the tree is 64 feet. find the distance, x, from tom to the base of the tree. round your answer to the nearest tenth of a foot.

Explanation:

Step1: Find triangle's top angle

First, calculate the angle at the top of the triangle formed by Tom, the base of the tree, and the top of the tree. The tree makes a 78° angle with the ground, so its supplementary angle inside the triangle is $180^\circ - 78^\circ = 102^\circ$. The sum of angles in a triangle is $180^\circ$, so the top angle is:
$180^\circ - 30^\circ - 102^\circ = 48^\circ$

Step2: Apply Law of Sines

Use the Law of Sines, which states $\frac{x}{\sin(\text{top angle})} = \frac{\text{tree length}}{\sin(\text{elevation angle})}$. Substitute the known values:
$\frac{x}{\sin(48^\circ)} = \frac{64}{\sin(30^\circ)}$

Step3: Solve for x

Rearrange to solve for $x$, using $\sin(30^\circ) = 0.5$ and $\sin(48^\circ) \approx 0.7431$:
$x = \frac{64 \times \sin(48^\circ)}{\sin(30^\circ)} = \frac{64 \times 0.7431}{0.5} \approx 94.5$

Answer:

94.5 feet