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

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 587.37 ft. from the top of the cliff.
The climber is 587.37 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 326.42 ft. from the top of the cliff.

Answer 1

To determine how far the climber is from the top of the cliff, we can first find the height of the cliff using the angle of elevation of 65° and then find the height at which the climber is located using the angle of elevation of 40°.

1. **Height of the Cliff:**
- We have a right triangle where:
- The adjacent side (distance from the cliff) = 700 ft
- The angle of elevation to the top of the cliff = 65°

We can find the height of the cliff (h) using the tangent function:
\[
\tan(65°) = \frac{h}{700}
\]
Rearranging gives us:
\[
h = 700 \cdot \tan(65°)
\]

Using a calculator:
\[
h \approx 700 \cdot 2.1445 \approx 1501.15 \text{ ft}
\]

2. **Height of the Climber:**
- Now we can calculate the height of the climber using the angle of elevation of 40°:
\[
\tan(40°) = \frac{h_c}{700}
\]
Rearranging gives us:
\[
h_c = 700 \cdot \tan(40°)
\]

Using a calculator:
\[
h_c \approx 700 \cdot 0.8391 \approx 587.37 \text{ ft}
\]

3. **Distance from the Climbers to the Top of the Cliff:**
- Now we can find how far the climber is from the top of the cliff:
\[
\text{Distance from the top} = h - h_c
\]
\[
= 1501.15 \text{ ft} - 587.37 \text{ ft} \approx 913.78 \text{ ft}
\]

Therefore, the climber is approximately **913.78 ft from the top of the cliff**.

The correct response is:
**The climber is 913.78 ft. from the top of the cliff.**