Question
from the top of a vertical 40m high the angle of depression of an object that is level with the base of the cliff is 34 how far is the object from the base of the cliff the solution i use was tan (34)=40/x
x=40 tan (34)
x=40/0.6494=61.9m
61.9 meters
x=40 tan (34)
x=40/0.6494=61.9m
61.9 meters
Answers
GPT-4o mini
It looks like there’s a small misunderstanding in the setup of your equation for solving the problem. The angle of depression is measured from the horizontal line down to the object, so we need to use the correct trigonometric function.
Let's analyze the situation:
1. You have a vertical cliff that is 40 meters high.
2. The angle of depression from the top of the cliff to the object at the base is \(34^\circ\).
3. You want to find the horizontal distance from the base of the cliff to the object.
To solve this, we can use the tangent function, but we have to identify what triangle we are working with. Since the angle of depression is downwards to the object, we can think of this in terms of the right triangle formed by the height of the cliff, the horizontal distance to the object, and the line of sight from the top of the cliff to the object.
The tangent of the angle is given by the formula:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\theta\) is the angle of depression (34 degrees), the "opposite" side is the height of the cliff (40 m), and the "adjacent" side is the horizontal distance \(x\) from the base of the cliff to the object.
In your problem, we should set up the equation as follows:
\[
\tan(34^\circ) = \frac{40}{x}
\]
Now, we can rearrange the equation to solve for \(x\):
\[
x = \frac{40}{\tan(34^\circ)}
\]
Now, let's calculate \(x\) using the correct value for \(\tan(34^\circ)\).
\[
\tan(34^\circ) \approx 0.6745
\]
So,
\[
x = \frac{40}{0.6745} \approx 59.3 \, \text{meters}
\]
Thus, the horizontal distance from the base of the cliff to the object is approximately 59.3 meters.
Let's analyze the situation:
1. You have a vertical cliff that is 40 meters high.
2. The angle of depression from the top of the cliff to the object at the base is \(34^\circ\).
3. You want to find the horizontal distance from the base of the cliff to the object.
To solve this, we can use the tangent function, but we have to identify what triangle we are working with. Since the angle of depression is downwards to the object, we can think of this in terms of the right triangle formed by the height of the cliff, the horizontal distance to the object, and the line of sight from the top of the cliff to the object.
The tangent of the angle is given by the formula:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\theta\) is the angle of depression (34 degrees), the "opposite" side is the height of the cliff (40 m), and the "adjacent" side is the horizontal distance \(x\) from the base of the cliff to the object.
In your problem, we should set up the equation as follows:
\[
\tan(34^\circ) = \frac{40}{x}
\]
Now, we can rearrange the equation to solve for \(x\):
\[
x = \frac{40}{\tan(34^\circ)}
\]
Now, let's calculate \(x\) using the correct value for \(\tan(34^\circ)\).
\[
\tan(34^\circ) \approx 0.6745
\]
So,
\[
x = \frac{40}{0.6745} \approx 59.3 \, \text{meters}
\]
Thus, the horizontal distance from the base of the cliff to the object is approximately 59.3 meters.