An ant looks to the top of a bulding at an angle of elevation of 32°. The ant then walks an additional 66 feet from the edge of the building. If the new angle of elevation from the ant to the top of the building is 22°, find the height of the building. Round to the nearest tenth.

To solve this problem, we can use trigonometry. Let's break down the steps:

Step 1: Determine the height of the building based on the first angle of elevation.
We know that the angle of elevation is 32°. This means that if we consider the distance the ant walked as the base of a right triangle, and the height of the building as the opposite side, we have enough information to use the tangent function.

The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the building, and the adjacent side is the base (distance the ant walked).

So the equation becomes tan(32°) = height of building / 66. We can rearrange this equation to solve for the height of the building.

height of building = tan(32°) * 66

Using a scientific calculator, the height of the building is approximately 41.1 feet (rounded to the nearest tenth).

Step 2: Determine the new height of the building based on the second angle of elevation.
We now have a new angle of elevation of 22°. Similarly, we can set up another equation based on the tangent function:

tan(22°) = new height of building / 66

To find the new height of the building, we rearrange the equation:

new height of building = tan(22°) * 66

Using a scientific calculator, the new height of the building is approximately 30.8 feet (rounded to the nearest tenth).

Step 3: Calculate the height of the building.
To determine the height of the building, we need to find the difference between the new height and the original height:

height of the building = new height of building - height of building

height of the building = 30.8 - 41.1

height of the building = -10.3

Note that in this case, the result is negative because the original height (41.1) is greater than the new height (30.8). However, in the context of this problem, the height of a building cannot be negative.

Therefore, the height of the building is approximately 10.3 feet (rounded to the nearest tenth).

If the first look is from distance x, then

h/x = tan 32°
h/(x+66) = tan 22°

so, equating values for x, we have

h/tan32° = h/tan22° - 66
now just solve for h