Trigonometry questions

Question

To estimate the height of a building two students find the angle of elevation from a point (at general level) down the street from the building top of the building 30°. From a point that is 250 feet close to the building the angle of elevation (at ground level) to the top of the building is 56°. If we assume that the street is level, use this information to estimate the height of the building. The height of the building is _____ feet.

Answer 1

This one is a little tricky to explain without my paper for use. First, we have the first triangle. We know the angle from the ground is 30 degrees. We can call the height of the building itself x. We know that since the building is vertical and one of the angles is 30 degrees, we must have a 30-60-90 triangle. We know that if the height of the building is x, the length your friend is from the building (the side adjacent to the 30-degree angle) is √3x. We also know that the length of the side adjacent to the 56-degree angle is √3x - 250 since they are 250 feet closer to the building. Here is where trig comes in. We know that tan is opposite/adjacent. Since the adjacent side of the 56-degree angle is the √3x - 250 side, the opposite must be the height of the building, which is x. So we know that tan 56 = opposite/adjacent = x/(√3x-250). If we do some additional math, we get that (tan 56)*(√3x - 250) = x. Multiplying all the terms out we get, ((tan 56)*√3x) - (tan 56)*250 = x. Rearranging so all of the x's are on one side, we get ((tan 56)*√3x - x) = (tan 56)*250. This is equal to (((tan 56)*√3) - 1)x = (tan 56)*250. Plugging all the numbers, we get 1.5679x = 370.6402 which means x ≈ 236.3973.