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
55 feet
55 feet
66 feet
66 feet
680 inches
680 inches
(1 point)
Responses
240 inches
240 inches
55 feet
55 feet
66 feet
66 feet
680 inches
680 inches
Answers
GPT-4o mini
To solve for the height of the tree, we can use the given information and trigonometric ratios.
1. **Convert Shaina's height to inches:**
- Shaina's height: 5 feet 6 inches = \( (5 \times 12) + 6 = 60 + 6 = 66 \) inches
2. **Set up the scenario:**
- Shaina stands 20 feet from the base of the tree.
- The angle of elevation to the pileated woodpecker is 68 degrees.
3. **Use the tangent function:**
- The tangent of the angle of elevation (68 degrees) relates the height of the tree (h) above Shaina's eye level to the distance from Shaina to the tree (20 feet).
- Using the tangent function:
\[
\tan(68^\circ) = \frac{h}{20}
\]
4. **Calculate h:**
\[
h = 20 \cdot \tan(68^\circ)
\]
Using a calculator:
- Find \(\tan(68^\circ)\):
\[
\tan(68^\circ) \approx 2.4751
\]
Now calculate h:
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
5. **Add Shaina's height to the height above her eye level:**
The total height of the tree (H) is given by:
\[
H = h + \text{Shaina's height} = 49.502 \text{ feet} + 5.5 \text{ feet}
\]
Since Shaina's height in feet is 5.5 feet (6 inches):
\[
H \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
So we approximate the height of the tree, rounding where necessary, to obtain the final result.
The answer for the total height of the tree is approximately \( 55 \text{ feet} \).
**Final answer:**
- 55 feet
1. **Convert Shaina's height to inches:**
- Shaina's height: 5 feet 6 inches = \( (5 \times 12) + 6 = 60 + 6 = 66 \) inches
2. **Set up the scenario:**
- Shaina stands 20 feet from the base of the tree.
- The angle of elevation to the pileated woodpecker is 68 degrees.
3. **Use the tangent function:**
- The tangent of the angle of elevation (68 degrees) relates the height of the tree (h) above Shaina's eye level to the distance from Shaina to the tree (20 feet).
- Using the tangent function:
\[
\tan(68^\circ) = \frac{h}{20}
\]
4. **Calculate h:**
\[
h = 20 \cdot \tan(68^\circ)
\]
Using a calculator:
- Find \(\tan(68^\circ)\):
\[
\tan(68^\circ) \approx 2.4751
\]
Now calculate h:
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
5. **Add Shaina's height to the height above her eye level:**
The total height of the tree (H) is given by:
\[
H = h + \text{Shaina's height} = 49.502 \text{ feet} + 5.5 \text{ feet}
\]
Since Shaina's height in feet is 5.5 feet (6 inches):
\[
H \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
So we approximate the height of the tree, rounding where necessary, to obtain the final result.
The answer for the total height of the tree is approximately \( 55 \text{ feet} \).
**Final answer:**
- 55 feet