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

66 feet
66 feet

55 feet
55 feet

240 inches
240 inches

Answer 1

To find the height of the tree, we can use trigonometric ratios. We first need to determine Shaina's height in feet:

Shaina's height = 5 feet 6 inches = 5.5 feet.

Next, we know that Shaina is standing 20 feet from the base of the tree, and the angle of elevation to the top of the tree is 68 degrees. We can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of the right triangle formed.

Let \( h \) be the height of the tree above Shaina’s eye level. The tangent of the angle of elevation can be expressed as:

\[
\tan(68^\circ) = \frac{h}{20}
\]

We can rearrange this equation to solve for \( h \):

\[
h = 20 \cdot \tan(68^\circ)
\]

Now we calculate \( \tan(68^\circ) \):

\(\tan(68^\circ) \approx 2.4751\) (using a calculator)

Now substitute this value into the equation:

\[
h = 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]

Now, since this height \( h \) is above Shaina's height, we need to add her height to find the total height of the tree \( H \):

\[
H = h + 5.5 \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]

Rounding down, the total height of the tree is approximately 55 feet.

Therefore, the answer is:

**55 feet**