point A and B are 100m apart and are of the same as the foot of a building. the angle of elevation of the top of the building from point A an B are 21 degrees and 32 degree respectively. how far is A from the building?

make your sketch, and label the top of the building P and the bottom of the building R

In triangle ABP,
angle A = 21°
angle ABP = 148°
so angle APB = 53°

by the sine law:
BP/sin21 = 100/sin53
BP = 100sin21/sin53 = .... you do that, store answer in calculator's memory

in right-angled triangle BPR
cos 32 = BR/BP
BR = BPcos32

so AR = 100 + BR

just realized I made an arithmetic error in finding

angle APB

angle APB = 11° , not 53

so please make the necessary changes.

by the sine law:
BP/sin21 = 100/sin11
BP = 100sin21/sin11 = .... you do that, store answer in calculator's memory


the rest stays the same

To find the distance between point A and the building, we can use the concept of trigonometry and the given angles of elevation. Let's denote the distance from point A to the building as x.

We are given that point A and point B are 100m apart and are at the same level as the foot of the building. This forms a right triangle, where the base is the distance between points A and B (100 m) and the height is the distance from point A to the top of the building.

Let's analyze the right triangle formed by point A, the top of the building, and point B. In this triangle, we have the following information:

1. Angle of elevation from point A: 21 degrees
This means that if we draw a line from point A to the top of the building, it will make an angle of 21 degrees with the horizontal line (the ground).

2. Angle of elevation from point B: 32 degrees
Similarly, if we draw a line from point B to the top of the building, it will make an angle of 32 degrees with the horizontal line.

Now, we can use trigonometry to find the distance from point A to the building (x). We will utilize the tangent function.

Using the tangent function for the angle of elevation from point A, we have:

tan(21 degrees) = height of the building / x

Since we know the height of the building is the same for both point A and point B (as they are at the same level), we can rewrite the equation as:

tan(21 degrees) = height of the building / 100 m

Similarly, using the tangent function for the angle of elevation from point B, we have:

tan(32 degrees) = height of the building / (100 + x) m

Now, we have a system of two equations with two unknowns, x (the distance from point A to the building) and the height of the building.

We can solve this system of equations to find the value of x, which will give us the distance from point A to the building.

Solving the system of equations is beyond the scope of this explanation, but you can use various methods such as substitution or elimination to find the value of x.

Once you find the value of x, that will be the distance from point A to the building.