The angle of elevation to the top of a building is found to be 5 degrees from the ground at a distance of 4 miles from the base of the building. Find the height of the building to 4 decimal places.
h = 4 mi * tan(5º)
= 4 * 5280 ft * tan(5º)
To find the height of the building, we can use trigonometry.
First, let's draw a diagram to visualize the problem.
A
/|
/ |
/ |x
/ |
/____|
B C
In this diagram, point A represents the top of the building, point B represents the base of the building, and point C represents the location where the angle of elevation is measured from. The height of the building is represented by "x".
We are given that the angle of elevation, angle BAC, is 5 degrees, and the distance from point C to point B is 4 miles.
When dealing with trigonometry and angles, it's important to work in the same unit of measurement. Since the distance is given in miles, we will keep the height of the building in miles as well.
To find the height of the building, we will use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the height of the building is the opposite side, and the distance from point C to point B is the adjacent side.
Using the tangent function, we have:
tan(5 degrees) = x / 4 miles
To solve for x (the height), we need to isolate it. We can do this by multiplying both sides of the equation by 4 miles:
4 miles * tan(5 degrees) = x
Using a calculator or a mathematical software, we can evaluate the tangent of 5 degrees:
4 miles * 0.0874886635 = x
x ≈ 0.349954654 miles
Therefore, the height of the building is approximately 0.35 miles (rounded to 4 decimal places).