The altitude of the cloud base is particularly important at airports. (The cloud base is where the lowest layer of clouds begins to form.) A bright spotlight is directed vertically upward. It creates a bright spot on the cloud base. From a horizontal distance of 1,150 feet away form the spotlight, the angle of elevation, θ, is measured to be 55°. Find the height of the cloud base (to the nearest foot).
We form a rt. triangle:
X = 1150Ft. = Hor. side.
Y = Ver. side = Ht of cloud base.
Z = Hyp. = Line of sight.
A = 55 Deg. = Angle bet. hyp. and gnd.
tanA = Y/X
Y = X*tanA = 1150*tann55 = 1642 Ft.
To find the height of the cloud base, we can use trigonometry and the concept of similar triangles.
Let's start by drawing a diagram. We have a spotlight at the bottom and a cloud base at some height. The distance from the spotlight to the cloud base is the height we are trying to find.
/
/
/
/
/
/θ
/|
/ |
/ |
/ |
/ |height of cloud base
/ |
/______|
spotlight
First, let's label the given information:
The angle of elevation θ = 55°.
The horizontal distance from the spotlight to the cloud base = 1,150 feet.
Now, let's use trigonometry to relate the angle θ and the height of the cloud base.
The tangent of an angle is defined as the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the cloud base, and the adjacent side is the horizontal distance from the spotlight.
Using the tangent function, we can set up the following equation:
tan(θ) = height of cloud base / horizontal distance
Substituting the given values:
tan(55°) = height of cloud base / 1,150 feet
Now, we can solve for the height of the cloud base.
height of cloud base = tan(55°) * 1,150 feet
Calculating this value:
height of cloud base ≈ 1.428 * 1,150 feet
height of cloud base ≈ 1,642.2 feet
Therefore, the height of the cloud base is approximately 1,642 feet (rounded to the nearest foot).