Question
Question
A rescue team is 700 ft. away from a vertical cliff. The team measures the angle of elevation to the top of the cliff as 65°. A climber is stranded on the cliff at an angle of elevation of 40°. How far is the climber from the top of the cliff?(1 point)
Responses which answer is correct and why
The climber is 1,501.15 ft. from the top of the cliff.
The climber is 1,501.15 ft. from the top of the cliff.
The climber is 913.78 ft. from the top of the cliff.
The climber is 913.78 ft. from the top of the cliff.
The climber is 587.37 ft. from the top of the cliff.
The climber is 587.37 ft. from the top of the cliff.
The climber is 326.42 ft. from the top of the cliff.
The climber is 326.42 ft. from the top of the cliff.
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A rescue team is 700 ft. away from a vertical cliff. The team measures the angle of elevation to the top of the cliff as 65°. A climber is stranded on the cliff at an angle of elevation of 40°. How far is the climber from the top of the cliff?(1 point)
Responses which answer is correct and why
The climber is 1,501.15 ft. from the top of the cliff.
The climber is 1,501.15 ft. from the top of the cliff.
The climber is 913.78 ft. from the top of the cliff.
The climber is 913.78 ft. from the top of the cliff.
The climber is 587.37 ft. from the top of the cliff.
The climber is 587.37 ft. from the top of the cliff.
The climber is 326.42 ft. from the top of the cliff.
The climber is 326.42 ft. from the top of the cliff.
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Answers
GPT-4o mini
To solve this problem, we will first calculate the height of the cliff, and then determine the height of the climber above the ground, from which we can find how far the climber is from the top of the cliff.
1. **Calculate the height of the cliff (H):**
- The angle of elevation to the top of the cliff is 65°.
- The distance from the rescue team to the cliff is 700 ft.
- Using the tangent of the angle:
\[
\tan(65°) = \frac{H}{700}
\]
Rearranging gives:
\[
H = 700 \cdot \tan(65°)
\]
Using a calculator:
\[
H \approx 700 \cdot 2.1445 \approx 1501.15 \text{ ft}
\]
2. **Calculate the height of the climber (h):**
- The angle of elevation to the climber is 40°.
- Using the tangent again:
\[
\tan(40°) = \frac{h}{700}
\]
Rearranging gives:
\[
h = 700 \cdot \tan(40°)
\]
Calculating:
\[
h \approx 700 \cdot 0.8391 \approx 587.37 \text{ ft}
\]
3. **Calculate the distance from the top of the cliff to the climber:**
- The distance from the top of the cliff to the climber is the height of the cliff minus the height of the climber:
\[
\text{Distance from top to climber} = H - h = 1501.15 \text{ ft} - 587.37 \text{ ft} \approx 913.78 \text{ ft}
\]
Therefore, the climber is approximately **913.78 ft. from the top of the cliff**.
The correct answer is: **The climber is 913.78 ft. from the top of the cliff.**
1. **Calculate the height of the cliff (H):**
- The angle of elevation to the top of the cliff is 65°.
- The distance from the rescue team to the cliff is 700 ft.
- Using the tangent of the angle:
\[
\tan(65°) = \frac{H}{700}
\]
Rearranging gives:
\[
H = 700 \cdot \tan(65°)
\]
Using a calculator:
\[
H \approx 700 \cdot 2.1445 \approx 1501.15 \text{ ft}
\]
2. **Calculate the height of the climber (h):**
- The angle of elevation to the climber is 40°.
- Using the tangent again:
\[
\tan(40°) = \frac{h}{700}
\]
Rearranging gives:
\[
h = 700 \cdot \tan(40°)
\]
Calculating:
\[
h \approx 700 \cdot 0.8391 \approx 587.37 \text{ ft}
\]
3. **Calculate the distance from the top of the cliff to the climber:**
- The distance from the top of the cliff to the climber is the height of the cliff minus the height of the climber:
\[
\text{Distance from top to climber} = H - h = 1501.15 \text{ ft} - 587.37 \text{ ft} \approx 913.78 \text{ ft}
\]
Therefore, the climber is approximately **913.78 ft. from the top of the cliff**.
The correct answer is: **The climber is 913.78 ft. from the top of the cliff.**