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Use trigonometric ratios to solve the following problem. Shaina, who is 5 feet, 6 inches tall, is standing 20 feet from the bas...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 680 inches 680 inches 66 feet 66 feet 55 feet 55 feet 240 inches
Answers
GPT-4o mini
To determine the height of the tree with respect to Shaina's height, we can use the tangent function, which relates angles and dimensions in a right triangle.
1. Convert Shaina's height to inches:
\[
\text{Height of Shaina} = 5 \text{ feet} \times 12 \text{ inches/foot} + 6 \text{ inches} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. The distance from Shaina to the base of the tree is given as 20 feet, which we convert to inches:
\[
\text{Distance to tree} = 20 \text{ feet} \times 12 \text{ inches/foot} = 240 \text{ inches}
\]
3. The angle of elevation to the woodpecker is given as 68 degrees. We can set up the equation using the tangent of the angle:
\[
\tan(68^\circ) = \frac{\text{height of the tree above Shaina}}{\text{distance to tree}}
\]
Let \( h \) be the height of the tree above Shaina, then:
\[
\tan(68^\circ) = \frac{h}{240}
\]
4. Now we solve for \( h \):
\[
h = 240 \cdot \tan(68^\circ)
\]
5. To calculate \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751 \quad \text{(using a scientific calculator)}
\]
Thus:
\[
h \approx 240 \cdot 2.4751 \approx 594.24 \text{ inches}
\]
6. Add Shaina's height to find the total height of the tree:
\[
\text{Height of the tree} = h + \text{Height of Shaina} = 594.24 + 66 = 660.24 \text{ inches}
\]
7. Convert to feet if required:
\[
\text{Height in feet} = \frac{660.24}{12} \approx 55.02 \text{ feet}
\]
In conclusion, rounding to the nearest whole number, the height of the tree is approximately \( 55 \) feet.
Thus, the correct response is **55 feet**.
1. Convert Shaina's height to inches:
\[
\text{Height of Shaina} = 5 \text{ feet} \times 12 \text{ inches/foot} + 6 \text{ inches} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches}
\]
2. The distance from Shaina to the base of the tree is given as 20 feet, which we convert to inches:
\[
\text{Distance to tree} = 20 \text{ feet} \times 12 \text{ inches/foot} = 240 \text{ inches}
\]
3. The angle of elevation to the woodpecker is given as 68 degrees. We can set up the equation using the tangent of the angle:
\[
\tan(68^\circ) = \frac{\text{height of the tree above Shaina}}{\text{distance to tree}}
\]
Let \( h \) be the height of the tree above Shaina, then:
\[
\tan(68^\circ) = \frac{h}{240}
\]
4. Now we solve for \( h \):
\[
h = 240 \cdot \tan(68^\circ)
\]
5. To calculate \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751 \quad \text{(using a scientific calculator)}
\]
Thus:
\[
h \approx 240 \cdot 2.4751 \approx 594.24 \text{ inches}
\]
6. Add Shaina's height to find the total height of the tree:
\[
\text{Height of the tree} = h + \text{Height of Shaina} = 594.24 + 66 = 660.24 \text{ inches}
\]
7. Convert to feet if required:
\[
\text{Height in feet} = \frac{660.24}{12} \approx 55.02 \text{ feet}
\]
In conclusion, rounding to the nearest whole number, the height of the tree is approximately \( 55 \) feet.
Thus, the correct response is **55 feet**.