Question
A rock climber is scaling a wall with an angle of 72° from the ground. a. If she is 241 feet off the ground, what distance on the wall has she traveled? b. The belay, or person handling her ropes from the ground, is standing 50 feet from the base of the wall. What is the angle of elevation between the belay and the climber? Recall that the wall is slanted.
Answers
GPT-4o mini
To solve the problem, we can make use of trigonometric relationships.
### Part a: Finding the distance traveled on the wall
Given that the wall is at an angle of \(72^\circ\) from the ground and the climber is \(241\) feet off the ground, we can use the sine function to find the distance traveled along the wall (\(d\)).
The relationship is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
where \(\theta\) is the angle of elevation (72°), the "opposite" is the height (241 feet), and the "hypotenuse" is the distance traveled on the wall (d).
So, we have:
\[
\sin(72^\circ) = \frac{241}{d}
\]
To find \(d\):
\[
d = \frac{241}{\sin(72^\circ)}
\]
Calculating \(\sin(72^\circ)\):
\[
\sin(72^\circ) \approx 0.9511
\]
Now, substituting the values:
\[
d = \frac{241}{0.9511} \approx 253.6 \text{ feet}
\]
### Part b: Finding the angle of elevation between the belay and the climber
To find the angle of elevation (\(\phi\)) from the belay to the climber, we can use the tangent function, where:
\[
\tan(\phi) = \frac{\text{opposite}}{\text{adjacent}}
\]
In our case, the "opposite" is the height of the climber above the ground (241 feet) and the "adjacent" is the horizontal distance from the base of the wall to the belay (50 feet).
So,
\[
\tan(\phi) = \frac{241}{50}
\]
Calculating \(\tan(\phi)\):
\[
\tan(\phi) \approx 4.82
\]
Now, we can find the angle \(\phi\) by taking the arctangent:
\[
\phi = \tan^{-1}(4.82)
\]
Calculating \(\phi\):
\[
\phi \approx 78.57^\circ
\]
### Summary of results:
a. The distance traveled on the wall is approximately \(253.6\) feet.
b. The angle of elevation from the belay to the climber is approximately \(78.57^\circ\).
### Part a: Finding the distance traveled on the wall
Given that the wall is at an angle of \(72^\circ\) from the ground and the climber is \(241\) feet off the ground, we can use the sine function to find the distance traveled along the wall (\(d\)).
The relationship is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
where \(\theta\) is the angle of elevation (72°), the "opposite" is the height (241 feet), and the "hypotenuse" is the distance traveled on the wall (d).
So, we have:
\[
\sin(72^\circ) = \frac{241}{d}
\]
To find \(d\):
\[
d = \frac{241}{\sin(72^\circ)}
\]
Calculating \(\sin(72^\circ)\):
\[
\sin(72^\circ) \approx 0.9511
\]
Now, substituting the values:
\[
d = \frac{241}{0.9511} \approx 253.6 \text{ feet}
\]
### Part b: Finding the angle of elevation between the belay and the climber
To find the angle of elevation (\(\phi\)) from the belay to the climber, we can use the tangent function, where:
\[
\tan(\phi) = \frac{\text{opposite}}{\text{adjacent}}
\]
In our case, the "opposite" is the height of the climber above the ground (241 feet) and the "adjacent" is the horizontal distance from the base of the wall to the belay (50 feet).
So,
\[
\tan(\phi) = \frac{241}{50}
\]
Calculating \(\tan(\phi)\):
\[
\tan(\phi) \approx 4.82
\]
Now, we can find the angle \(\phi\) by taking the arctangent:
\[
\phi = \tan^{-1}(4.82)
\]
Calculating \(\phi\):
\[
\phi \approx 78.57^\circ
\]
### Summary of results:
a. The distance traveled on the wall is approximately \(253.6\) feet.
b. The angle of elevation from the belay to the climber is approximately \(78.57^\circ\).