a boy standing on a vertical cliff 50m high is looking down an angle of depression of 20 degree at a car. how far is the car from the base of the cliff?
To solve this problem, we can use trigonometry. Let's break down the problem into two parts: the height of the cliff and the distance of the car from the base of the cliff.
1. Finding the height of the cliff:
The angle of depression is the angle between the horizontal line and the line of sight from the boy to the car. In this case, it is given as 20 degrees. Since the boy is standing on a vertical cliff, the angle between the vertical line and the line of sight is 90 degrees.
Using trigonometry, we can use the tangent function to find the height of the cliff:
tan(angle of depression) = height of the cliff / distance to the car
tan(20 degrees) = 50m / distance to the car
To find the height of the cliff, we rearrange the equation:
50m = distance to the car * tan(20 degrees)
2. Finding the distance of the car from the base of the cliff:
Now, we need to find the distance of the car from the base of the cliff. We can use the same equation as before, but rearrange it to solve for the distance to the car:
distance to the car = 50m / tan(20 degrees)
Using a calculator, we can find the value of tan(20 degrees) to be approximately 0.36397.
Substituting this value back into the equation, we can solve for the distance to the car:
distance to the car = 50m / 0.36397
distance to the car ≈ 137.32m
Therefore, the car is approximately 137.32 meters from the base of the cliff.
To find the distance between the car and the base of the cliff, we can use trigonometry.
We have an angle of depression of 20 degrees, which means the angle between the line of sight from the boy to the car and the horizontal ground is 20 degrees.
Let's denote the distance from the base of the cliff to the car as 'x' (in meters).
In this scenario, the height of the cliff acts as the opposite side of a right triangle, the distance from the base of the cliff to the car acts as the adjacent side, and the angle of depression is the angle between the hypotenuse and the adjacent side.
Using the tangent function, we can calculate the distance 'x' as follows:
tan(20 degrees) = opposite/adjacent
tan(20 degrees) = 50/x
To solve for 'x', we can rearrange the equation:
x = 50 / tan(20 degrees)
Now, we can use a calculator to find the value of 'x'.
x ≈ 143.33 meters
Therefore, the car is approximately 143.33 meters away from the base of the cliff.
Tan20 = h/d = 50/d
d = 50/Tan20 = 137.4 m.