A man is standing 35 m from a building. On top of the building is a flg. From the man, the angle of elevation of the top of the building is pi/6 radians and the angle of elevation of the top of the flag is pi/3 radians. Determine an exact expression for the distance from the top of the flag to the top of the building.

Question

A man is standing 35 m from a building. On top of the building is a flg. From the man, the angle of elevation of the top of the building is pi/6 radians and the angle of elevation of the top of the flag is pi/3 radians. Determine an exact expression for the distance from the top of the flag to the top of the building.

Answer 1

Did you make a diagram?

I see it as
height of flag above building
= 35tan60º - 35tan30º

Answer 2

To determine the distance from the top of the flag to the top of the building, we can use trigonometry. Let's denote the distance from the man to the top of the building as 'x' and the distance from the top of the building to the top of the flag as 'd'.

Based on the given information, we have two right-angled triangles. The first triangle consists of the man, the top of the building, and the ground, while the second triangle consists of the top of the building, the top of the flag, and a perpendicular line dropped from the top of the flag to the ground.

Let's consider the first triangle. The angle of elevation of the top of the building from the man is pi/6 radians. Since we know that the adjacent side of an angle in a right triangle is equal to the distance from the observer (in this case, the man) to the object (the top of the building), we can say:

Adjacent side = x

Now, let's consider the second triangle. The angle of elevation of the top of the flag from the man is pi/3 radians. Since we know that the adjacent side of an angle in a right triangle is equal to the distance from the observer (in this case, the top of the building) to the object (the top of the flag), we can say:

Adjacent side = d

In both triangles, the opposite side is the height of the building (h). We are not given its value, but we can use the information to solve for it.

Using trigonometry, we can write the following relationships:

tan(pi/6) = h / x (1)
tan(pi/3) = h / d (2)

Now, we need to solve these equations to find the value of d.

From equation (1), we have:

h = x * tan(pi/6)

Substituting this value of h into equation (2), we get:

tan(pi/3) = (x * tan(pi/6)) / d

Now we can solve for d:

d = (x * tan(pi/6)) / tan(pi/3)

Simplifying this expression gives us the exact distance from the top of the flag to the top of the building.