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
Answers
GPT-4o mini
To determine the height of the tree, we first need to find the height of Shaina and then apply the trigonometric ratio given the angle of elevation and her distance from the tree.
1. **Convert Shaina's height to inches:**
- Shaina's height is 5 feet 6 inches.
- There are 12 inches in a foot, so:
\[
5 \text{ feet} = 5 \times 12 = 60 \text{ inches}
\]
\[
\text{Total height} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. **Identify the distance from Shaina to the tree:**
- This distance is given as 20 feet. We need to convert this to inches as well:
\[
20 \text{ feet} = 20 \times 12 = 240 \text{ inches}
\]
3. **Use the angle of elevation to find the height from Shaina's eyes to the top of the tree:**
- The angle of elevation is 68 degrees.
- We will use the tangent function, which relates the angle to the opposite side (height from Shaina's eye level to the top of the tree) over the adjacent side (distance from Shaina to the tree). Let \( h \) be the height from Shaina's eyes to the bird at the top of the tree:
\[
\tan(68^\circ) = \frac{h}{240 \text{ inches}}
\]
Rearranging gives:
\[
h = 240 \times \tan(68^\circ)
\]
4. **Calculate \( h \):**
- You can use a calculator to find \( \tan(68^\circ) \). The approximate value is:
\[
\tan(68^\circ) \approx 2.4751
\]
Thus,
\[
h \approx 240 \times 2.4751 \approx 594.2 \text{ inches}
\]
5. **Calculate the total height of the tree:**
- To find the total height of the tree, we need to add Shaina's height (in inches) to \( h \):
\[
\text{Total height of tree} = h + \text{Shaina's height}
\]
\[
= 594.2 + 66 = 660.2 \text{ inches}
\]
6. **Convert to feet if necessary:**
- Converting inches back to feet:
\[
\frac{660.2}{12} \approx 55.02 \text{ feet}
\]
The height of the tree is approximately **55 feet**. Thus, the correct response is:
**55 feet.**
1. **Convert Shaina's height to inches:**
- Shaina's height is 5 feet 6 inches.
- There are 12 inches in a foot, so:
\[
5 \text{ feet} = 5 \times 12 = 60 \text{ inches}
\]
\[
\text{Total height} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. **Identify the distance from Shaina to the tree:**
- This distance is given as 20 feet. We need to convert this to inches as well:
\[
20 \text{ feet} = 20 \times 12 = 240 \text{ inches}
\]
3. **Use the angle of elevation to find the height from Shaina's eyes to the top of the tree:**
- The angle of elevation is 68 degrees.
- We will use the tangent function, which relates the angle to the opposite side (height from Shaina's eye level to the top of the tree) over the adjacent side (distance from Shaina to the tree). Let \( h \) be the height from Shaina's eyes to the bird at the top of the tree:
\[
\tan(68^\circ) = \frac{h}{240 \text{ inches}}
\]
Rearranging gives:
\[
h = 240 \times \tan(68^\circ)
\]
4. **Calculate \( h \):**
- You can use a calculator to find \( \tan(68^\circ) \). The approximate value is:
\[
\tan(68^\circ) \approx 2.4751
\]
Thus,
\[
h \approx 240 \times 2.4751 \approx 594.2 \text{ inches}
\]
5. **Calculate the total height of the tree:**
- To find the total height of the tree, we need to add Shaina's height (in inches) to \( h \):
\[
\text{Total height of tree} = h + \text{Shaina's height}
\]
\[
= 594.2 + 66 = 660.2 \text{ inches}
\]
6. **Convert to feet if necessary:**
- Converting inches back to feet:
\[
\frac{660.2}{12} \approx 55.02 \text{ feet}
\]
The height of the tree is approximately **55 feet**. Thus, the correct response is:
**55 feet.**