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?
680 inches
55 feet
66 feet
240 inches
680 inches
55 feet
66 feet
240 inches
Answers
GPT-4o mini
To find the height of the tree in this scenario, we will break it down step by step using trigonometric ratios.
1. **Convert Shaina's height to inches:**
Shaina is 5 feet 6 inches tall. First, we convert this to inches.
\[
5 \text{ feet} = 5 \times 12 = 60 \text{ inches}
\]
\[
\text{Total height of Shaina} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. **Set up the scenario:**
Let \( h_T \) be the height of the tree in inches. Since Shaina is 20 feet away from the base of the tree, we convert that distance to inches:
\[
20 \text{ feet} = 20 \times 12 = 240 \text{ inches}
\]
3. **Form a right triangle:**
We can visualize a right triangle where:
- The distance from Shaina to the tree is the base (240 inches).
- The height from Shaina's eyes to the top of the tree is the vertical side.
- The angle of elevation from Shaina's eyes to the top of the tree is 68 degrees.
4. **Use the tangent function:**
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case:
\[
\tan(68^\circ) = \frac{\text{Height above Shaina's head}}{\text{Distance from tree}} = \frac{h_T - 66}{240}
\]
5. **Rearranging the formula:**
To find the height above Shaina's head, we can rearrange:
\[
h_T - 66 = 240 \tan(68^\circ)
\]
\[
h_T = 240 \tan(68^\circ) + 66
\]
6. **Calculate \( \tan(68^\circ) \):**
Using a calculator, we find:
\[
\tan(68^\circ) \approx 2.4751
\]
7. **Substituting back to find height:**
Now substituting this value back:
\[
h_T = 240 \times 2.4751 + 66
\]
\[
h_T \approx 594.24 + 66 = 660.24 \text{ inches}
\]
8. **Converting to feet:**
To convert inches back to feet:
\[
h_T \approx 660.24 \text{ inches} \div 12 \approx 55.02 \text{ feet}
\]
Based on the options provided, the closest match to our result is **55 feet**.
Thus, the height of the tree is approximately **55 feet**.
1. **Convert Shaina's height to inches:**
Shaina is 5 feet 6 inches tall. First, we convert this to inches.
\[
5 \text{ feet} = 5 \times 12 = 60 \text{ inches}
\]
\[
\text{Total height of Shaina} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. **Set up the scenario:**
Let \( h_T \) be the height of the tree in inches. Since Shaina is 20 feet away from the base of the tree, we convert that distance to inches:
\[
20 \text{ feet} = 20 \times 12 = 240 \text{ inches}
\]
3. **Form a right triangle:**
We can visualize a right triangle where:
- The distance from Shaina to the tree is the base (240 inches).
- The height from Shaina's eyes to the top of the tree is the vertical side.
- The angle of elevation from Shaina's eyes to the top of the tree is 68 degrees.
4. **Use the tangent function:**
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case:
\[
\tan(68^\circ) = \frac{\text{Height above Shaina's head}}{\text{Distance from tree}} = \frac{h_T - 66}{240}
\]
5. **Rearranging the formula:**
To find the height above Shaina's head, we can rearrange:
\[
h_T - 66 = 240 \tan(68^\circ)
\]
\[
h_T = 240 \tan(68^\circ) + 66
\]
6. **Calculate \( \tan(68^\circ) \):**
Using a calculator, we find:
\[
\tan(68^\circ) \approx 2.4751
\]
7. **Substituting back to find height:**
Now substituting this value back:
\[
h_T = 240 \times 2.4751 + 66
\]
\[
h_T \approx 594.24 + 66 = 660.24 \text{ inches}
\]
8. **Converting to feet:**
To convert inches back to feet:
\[
h_T \approx 660.24 \text{ inches} \div 12 \approx 55.02 \text{ feet}
\]
Based on the options provided, the closest match to our result is **55 feet**.
Thus, the height of the tree is approximately **55 feet**.