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 degrees. A climber is stranded on the cliff at an angle of elevation of 40 degrees. How far is the climber from the top of the cliff?
The climber is 326.42 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 587.37 ft from the top of the cliff.
The climber is 326.42 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 587.37 ft from the top of the cliff.
Answers
GPT-4o mini
To solve this, we can use trigonometry to determine the height of the cliff and the height of the climber based on the angles of elevation provided.
**Step 1: Calculate the height of the cliff.**
The angle of elevation to the top of the cliff is 65 degrees, and the distance from the base of the cliff to the rescue team is 700 ft. We will use the tangent function, which is defined as:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where the opposite is the height of the cliff (h) and the adjacent is the distance from the team to the cliff (700 ft).
So, we have:
\[
\tan(65^\circ) = \frac{h}{700}
\]
Calculating \( h \):
\[
h = 700 \cdot \tan(65^\circ)
\]
Using a calculator to find \( \tan(65^\circ) \):
\[
\tan(65^\circ) \approx 2.1445
\]
Therefore,
\[
h = 700 \cdot 2.1445 \approx 1501.15 \text{ ft}
\]
So, the height of the cliff is approximately 1501.15 ft.
**Step 2: Calculate the height of the climber.**
The angle of elevation to the climber is 40 degrees. We will use the same adjacent distance (700 ft) to calculate the height from the ground to the climber (let's call it \( h_c \)):
Using the tangent function again:
\[
\tan(40^\circ) = \frac{h_c}{700}
\]
Calculating \( h_c \):
\[
h_c = 700 \cdot \tan(40^\circ)
\]
Finding \( \tan(40^\circ) \):
\[
\tan(40^\circ) \approx 0.8391
\]
Thus,
\[
h_c = 700 \cdot 0.8391 \approx 587.37 \text{ ft}
\]
**Step 3: Calculate the distance from the climber to the top of the cliff.**
To find how far the climber is from the top of the cliff:
\[
\text{Distance from climber to top} = h - h_c
\]
Where \( h \) is the total height of the cliff, and \( h_c \) is the height of the climber:
\[
\text{Distance from climber to top} = 1501.15 - 587.37 \approx 913.78 \text{ ft}
\]
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.**
**Step 1: Calculate the height of the cliff.**
The angle of elevation to the top of the cliff is 65 degrees, and the distance from the base of the cliff to the rescue team is 700 ft. We will use the tangent function, which is defined as:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where the opposite is the height of the cliff (h) and the adjacent is the distance from the team to the cliff (700 ft).
So, we have:
\[
\tan(65^\circ) = \frac{h}{700}
\]
Calculating \( h \):
\[
h = 700 \cdot \tan(65^\circ)
\]
Using a calculator to find \( \tan(65^\circ) \):
\[
\tan(65^\circ) \approx 2.1445
\]
Therefore,
\[
h = 700 \cdot 2.1445 \approx 1501.15 \text{ ft}
\]
So, the height of the cliff is approximately 1501.15 ft.
**Step 2: Calculate the height of the climber.**
The angle of elevation to the climber is 40 degrees. We will use the same adjacent distance (700 ft) to calculate the height from the ground to the climber (let's call it \( h_c \)):
Using the tangent function again:
\[
\tan(40^\circ) = \frac{h_c}{700}
\]
Calculating \( h_c \):
\[
h_c = 700 \cdot \tan(40^\circ)
\]
Finding \( \tan(40^\circ) \):
\[
\tan(40^\circ) \approx 0.8391
\]
Thus,
\[
h_c = 700 \cdot 0.8391 \approx 587.37 \text{ ft}
\]
**Step 3: Calculate the distance from the climber to the top of the cliff.**
To find how far the climber is from the top of the cliff:
\[
\text{Distance from climber to top} = h - h_c
\]
Where \( h \) is the total height of the cliff, and \( h_c \) is the height of the climber:
\[
\text{Distance from climber to top} = 1501.15 - 587.37 \approx 913.78 \text{ ft}
\]
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.**