An antenna is located at the top of a building 1000 ft tall. From a point on the same horizontal plane as the base of the building, the angles of elevation from the top and the bottom of the antenna are 65.8 degrees and 62.6 degrees respectively. How tall is the antenna.

Thanks

Make a sketch, I have 2 right-angled triangles

Let the point of observation be x ft from the building, let the height of the antenna be h ft
then
tan 62.2 =100/x
x = 100/tan62.2 = 52.724... (I stored in memory to keep accuracy)

In 2nd triangle
tan 65.8 = (h+100)/x
h+100 = xtan65.8
h = 52.724(tan65.8) - 100 = 17.32 ft

To find the height of the antenna, we can use trigonometry and the concept of similar triangles.

Let's denote the height of the antenna as "h" (the unknown we want to find).

From the given information, we are given two angles of elevation: 65.8 degrees and 62.6 degrees. These angles are measured from a point on the same horizontal plane as the base of the building.

Now, let’s consider the two right triangles formed by the top and bottom of the antenna with the point on the ground:

Triangle 1: This triangle consists of the part of the building from the base to the bottom of the antenna. The angle of elevation of this triangle is 62.6 degrees.

Triangle 2: This triangle consists of the entire building from the base to the top of the antenna. The angle of elevation of this triangle is 65.8 degrees.

Now, the key observation is that these two triangles share the same base (the part of the building from the base to the bottom of the antenna) and have the same vertical angles.

Using the concept of similar triangles, we know that the ratios of corresponding sides in two similar triangles are equal.

Let's denote the length of the shared base of the triangles (the part of the building from the base to the bottom of the antenna) as "x".

In Triangle 1:
tan(62.6 degrees) = h / x

In Triangle 2:
tan(65.8 degrees) = (h + 1000) / x

Now we have two equations with two unknowns (h and x). We can solve these equations simultaneously to find the values of h and x.

To simplify the equations, we can take the tangent of both sides:

tan(62.6 degrees) = h / x
tan(65.8 degrees) = (h + 1000) / x

Now we can substitute the values of the tangent of the angles:

0.8746 = h / x
2.1885 = (h + 1000) / x

We can solve these two equations simultaneously to find the values of h and x.

From the first equation, we can isolate h:

h = 0.8746 * x

Now substitute this value of h in the second equation:

2.1885 = (0.8746 * x + 1000) / x

Multiply both sides of the equation by x to eliminate the denominator:

2.1885x = 0.8746x + 1000

Subtract 0.8746x from both sides:

1.3139x = 1000

Divide both sides by 1.3139 to solve for x:

x = 1000 / 1.3139

x ≈ 760.33 ft

Now plug the value of x back into the first equation to find h:

h = 0.8746 * 760.33

h ≈ 665.28 ft

Therefore, the height of the antenna is approximately 665.28 ft.