Question

Use the image to answer the question.

A grid shows a right triangle upper A upper B upper C. The angle of the vertex upper B is at the right angle.

Use trigonometric ratios to solve the following problem. Shaina, who is 5 feet, 6 inches tall, is standing 20 feet from the base of a tree when she sees a pileated woodpecker at the top of the tree. The bird is at an angle of elevation of 68 degrees above Shaina’s line of sight. What is the height of the tree?

(1 point)
Responses

680 inches
680 inches

55 feet
55 feet

240 inches
240 inches

66 feet
66 feet

Answer 1

To solve the problem, we need to determine the height of the tree using the angle of elevation and Shaina's height.

First, we convert Shaina's height from feet and inches to inches only:

- Shaina's height: 5 feet 6 inches = 5 * 12 + 6 = 66 inches.

Next, let's define:

- Let \( h \) be the height of the tree.
- The distance from Shaina to the base of the tree is 20 feet = \( 20 \times 12 = 240 \) inches.
- The angle of elevation to the woodpecker is 68 degrees.

Using the angle of elevation, we can form a right triangle where:

- The vertical side (opposite to the angle) is the height of the woodpecker above Shaina's head.
- The horizontal side (adjacent to the angle) is the distance from Shaina to the base of the tree.

We find the height above Shaina's head using the tangent function, which relates the opposite side to the adjacent side for an angle in a right triangle:

\[
\tan(68^\circ) = \frac{h - 66}{240}
\]

Here, \( h - 66 \) is the height of the woodpecker above Shaina's head.

Now, solving for \( h \):

1. Rearrange the equation:
\[
h - 66 = 240 \tan(68^\circ)
\]

2. Calculate \( \tan(68^\circ) \):
Using a calculator, \( \tan(68^\circ) \approx 2.475 \).

3. Substitute back into the equation:
\[
h - 66 = 240 \times 2.475 \approx 594
\]

4. Solve for \( h \):
\[
h = 594 + 66 = 660 \text{ inches}
\]

Now, since the options are given in inches or feet, we can convert:

\[
660 \text{ inches} = \frac{660}{12} \approx 55 \text{ feet}
\]

Thus, the height of the tree is:

**55 feet.**