Question
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 240 inches 240 inches 66 feet 66 feet 680 inches 680 inches 55 feet 55 feet
Answers
GPT-4o mini
To determine the height of the tree, we can use trigonometric ratios. First, we need to convert Shaina's height from feet and inches to just inches:
Shaina's height:
- 5 feet = 5 × 12 = 60 inches
- 6 inches = 6 inches
- Total height = 60 inches + 6 inches = 66 inches.
Now, we need to use the tangent of the angle of elevation to relate the height of the tree (let's call it \( h \)) to the distance from Shaina to the base of the tree.
The formula for the tangent of an angle in a right triangle is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case:
- \( \theta = 68^\circ \)
- The "opposite" side is the height of the tree above Shaina's eye level, which is \( h - 66 \) inches (since Shaina's eye level is at 66 inches).
- The "adjacent" side is the distance from Shaina to the tree, which is 20 feet. We need to convert that to inches since Shaina's height is in inches: \( 20 \text{ feet} = 20 \times 12 = 240 \text{ inches} \).
Putting this information into the formula gives:
\[
\tan(68^\circ) = \frac{h - 66}{240}
\]
Now, we can solve for \( h \):
\[
h - 66 = 240 \tan(68^\circ)
\]
Next, we need to calculate \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.475
\]
Now substitute this value back into the equation:
\[
h - 66 = 240 \times 2.475
\]
Now compute \( 240 \times 2.475 \):
\[
240 \times 2.475 \approx 594
\]
Now we add 66 inches to both sides to find \( h \):
\[
h = 594 + 66 = 660 \text{ inches}
\]
To convert this back to feet:
\[
660 \text{ inches} = \frac{660}{12} = 55 \text{ feet}
\]
Thus, the height of the tree is **55 feet**.
Therefore, the correct response is **55 feet**.
Shaina's height:
- 5 feet = 5 × 12 = 60 inches
- 6 inches = 6 inches
- Total height = 60 inches + 6 inches = 66 inches.
Now, we need to use the tangent of the angle of elevation to relate the height of the tree (let's call it \( h \)) to the distance from Shaina to the base of the tree.
The formula for the tangent of an angle in a right triangle is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case:
- \( \theta = 68^\circ \)
- The "opposite" side is the height of the tree above Shaina's eye level, which is \( h - 66 \) inches (since Shaina's eye level is at 66 inches).
- The "adjacent" side is the distance from Shaina to the tree, which is 20 feet. We need to convert that to inches since Shaina's height is in inches: \( 20 \text{ feet} = 20 \times 12 = 240 \text{ inches} \).
Putting this information into the formula gives:
\[
\tan(68^\circ) = \frac{h - 66}{240}
\]
Now, we can solve for \( h \):
\[
h - 66 = 240 \tan(68^\circ)
\]
Next, we need to calculate \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.475
\]
Now substitute this value back into the equation:
\[
h - 66 = 240 \times 2.475
\]
Now compute \( 240 \times 2.475 \):
\[
240 \times 2.475 \approx 594
\]
Now we add 66 inches to both sides to find \( h \):
\[
h = 594 + 66 = 660 \text{ inches}
\]
To convert this back to feet:
\[
660 \text{ inches} = \frac{660}{12} = 55 \text{ feet}
\]
Thus, the height of the tree is **55 feet**.
Therefore, the correct response is **55 feet**.