Questions: As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is 34°. If the point is 20 feet from the base of the tree, what is the height of the tree, to the nearest tenth of a foot? 1) 29.7 2) 16.6 3) 13.5 4) 11.2

As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is 34°.

If the point is 20 feet from the base of the tree, what is the height of the tree, to the nearest tenth of a foot?
1) 29.7
2) 16.6
3) 13.5
4) 11.2

Transcript text: As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is $34^{\circ}$. If the point is 20 feet from the base of the tree, what is the height of the tree, to the nearest tenth of a foot? 1) 29.7 2) 16.6 3) 13.5 4) 11.2

Solution

Solution Steps

Step 1: Identify the given information

The angle of elevation is 34°. The distance from the point on the ground to the base of the tree is 20 feet.

Step 2: Set up the trigonometric equation

Let $h$ be the height of the tree. We have a right triangle formed by the ground, the tree, and the line of sight. The angle of elevation is 34°, the adjacent side is 20 feet, and the opposite side is $h$. We can use the tangent function to relate these:

$\tan(34^\circ) = \frac{h}{20}$

Step 3: Solve for h

Multiply both sides by 20 to isolate $h$:

$h = 20 \tan(34^\circ)$

Using a calculator:

$h \approx 20 \times 0.6745 \approx 13.49$

Step 4: Round to the nearest tenth

Rounding 13.49 to the nearest tenth gives 13.5.