The angle of elevation to the top of a door is 47 degrees, and the angle of elevation to the ceiling above the door is 62 degrees. The ceiling is 10 feet above the floor. How tall is the door?

To calculate the door's height, we use the angles of elevation to the door and ceiling, and the known height of the ceiling to form two right-angled triangles. Using trigonometry, we solve for the height of the door using the tangent values of the angles provided.

Explanation:

To determine how tall the door is, we use the information about the angles of elevation and the height of the ceiling, which form two right-angled triangles with the door and the ceiling as their respective opposite sides.

Taking the angle of elevation to the top of the door as 47 degrees and the angle of elevation to the ceiling as 62 degrees with the ceiling being 10 feet off the floor, we can use trigonometry to find the height of the door.

First, we let x be the height of the door. The angle of elevation to the ceiling forms a larger triangle, and the door forms part of the opposite side of the smaller triangle with the same adjacent side (distance from the observer to the base of the door).

This means we can use the differences in the tangent values of these angles to solve for x.

Using the tangent of the angle of elevation:

tan(62°) = height of ceiling / base

base = height of ceiling / tan(62°)

tan(47°) = height of door / base

tan(47°) = x / (height of ceiling / tan(62°))

Thus, x (height of the door) = tan(47°) * (height of ceiling / tan(62°)). Plugging in the values, we get:

x = tan(47°) * (10 ft / tan(62°))

Carry out the calculation to find the height of the door.