How Do You Calculate the Height of a Building Using Trigonometry?

[illmatic899]

A surveyor stands on flat ground at an unknown distance from a tall building. She measures the angle from the horizonal ground to the top of the building; this angle is pi/3. next she paces 40ft further away from the building. the angle from the ground to the top of the building is now measured to be pi/4.
a)how tall is the building
b) If the surveyor moves 20 feet further from the building what will the angle from the horizontal to the building's roof be.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Let the height of the building, in feet, be h, the initial distance from the building be x. Then you have $tan(\pi/3)= \sqrt{3}= x/h$. 40 ft further away, the building is still h feet high and the distance from the building is now x+ 40 feet. Now you have $tan(\pi/4)= 1= (x+40)/h$. You now have two equations to solve for x and h.

For (b), let $\theta$. You have already solved for x so you know the distance from the building is x+ 40+ 20. And, of course, you have solved for h. Now, $tan(\theta)= (x+ 60)/h$. Solve that equation for $\theta$.

 

[Architeuthis Dux]

a) To solve for the height of the building, we can use the tangent function. We know that 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 distance from the surveyor to the building. So, we can set up the following equation: tan(pi/3) = height/x, where x is the unknown distance from the surveyor to the building. We can solve for x by using the tangent function and find that x = height/tan(pi/3). Similarly, when the surveyor moves 40ft further away, we can set up another equation: tan(pi/4) = height/(x+40). We can solve for x by using the tangent function and find that x = (height/tan(pi/4)) - 40. Now, we can set these two equations equal to each other and solve for the height of the building. So, (height/tan(pi/3)) = ((height/tan(pi/4)) - 40). We can solve for height by multiplying both sides by the common denominator, tan(pi/3) * tan(pi/4). This gives us height = (40*tan(pi/3)*tan(pi/4))/(tan(pi/4) - tan(pi/3)). Plugging this into a calculator, we find that the height of the building is approximately 40.8 feet.

b) If the surveyor moves 20 feet further from the building, the new distance from the surveyor to the building would be x+60. Using the same method as above, we can set up the following equation: tan(theta) = height/(x+60), where theta is the new angle from the ground to the top of the building. We can solve for theta by using the inverse tangent function and find that theta = arctan(height/(x+60)). Plugging in the value of height that we found in part a, we get theta = arctan(40.8/(x+60)). We can then plug in the value of x that we found in part a, which was x = (height/tan(pi/4)) - 40, and solve for theta. This gives us theta = arctan(40.8/((40.8/tan(pi/4)) - 40 + 60)). Plugging this into