To solve this problem, let's use trigonometric principles. Specifically, we'll use the tangent function, which relates the angle of elevation to the opposite side (height of the building) and the adjacent side (distance from the base of the building).
Given:
- The distance from the base of the building (adjacent side) is 50 feet.
- The angle of elevation is [tex]\( 60^\circ \)[/tex].
We need to find the height of the building (opposite side).
The tangent of an angle in a right triangle is defined as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, [tex]\(\theta = 60^\circ\)[/tex], the opposite is the height of the building (which we will call [tex]\( h \)[/tex]), and the adjacent side is 50 feet.
So, we write:
[tex]\[ \tan(60^\circ) = \frac{h}{50} \][/tex]
We know that [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex]. Substituting this value into the equation, we get:
[tex]\[ \sqrt{3} = \frac{h}{50} \][/tex]
To solve for [tex]\( h \)[/tex], we multiply both sides of the equation by 50:
[tex]\[ h = 50 \sqrt{3} \][/tex]
Thus, the height of the building is:
[tex]\[ 50 \sqrt{3} \, \text{feet} \][/tex]
So, the height of the building is [tex]\( \boxed{50 \sqrt{3}} \)[/tex] feet.
