A television antenna is on the roof of a building. From a point on the ground 36.0 ft from the building the angle of elevation of the top and bottom of the antenna are 51° 23' & 42°10', respectively. How tall is the antenna?

antenna height = h from roof up

building height = b
tan 42 10 = b/36 solve for b
then
tan 51 23 = (b+h) / 36

To find the height of the antenna, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.

Let's label the height of the antenna as "h" and the distance from the point on the ground to the building as "d". From the information given, we know that d = 36.0 ft.

Using the tangent function, we can set up two equations based on the angles of elevation:

tan(51° 23') = h / d
tan(42° 10') = (h + x) / d

We can solve for h by setting up an equation using these two equations:

h / d = tan(51° 23')
(h + x) / d = tan(42° 10')

Now, let's calculate the values:

First, we need to convert the angles from degrees and minutes to decimal form:

51° 23' = 51.3833°
42° 10' = 42.1667°

Now we can substitute the values into the equations:

h / 36.0 = tan(51.3833°)
(h + x) / 36.0 = tan(42.1667°)

Simplify the equations:

h = 36.0 * tan(51.3833°)
h + x = 36.0 * tan(42.1667°)

Now we can solve for h by subtracting the second equation from the first one:

h - (h + x) = 31.9696 - h

Simplifying the equation further gives us:

-h + x = -h + 25.8641

We can cancel out the "h" terms:

x = 25.8641

So, the height of the antenna is approximately 25.8641 ft.

To find the height of the antenna, we can use trigonometry. Let's break down the problem:

1. Start by drawing a diagram of the situation. Label the height of the antenna as 'h', the distance from the point on the ground to the building as 'x', and the angle of elevation to the top and bottom of the antenna as 'θ1' and 'θ2', respectively.

_____ <- Antenna top (height 'h')
| θ1 |
|______| <- Building
\
\
\ <- Angle of elevation 'θ2'
\
\
\
\
\ <- Distance from point on the ground to building (distance 'x')

2. Since we have the angle of elevation, we can use the tangent function. Recall that tangent(theta) is equal to opposite/adjacent, which in this case is the height 'h' divided by the distance 'x'. Therefore, we can write two equations:

tan(θ1) = h / x (equation 1)
tan(θ2) = (h + antenna height) / x (equation 2)

Note: We add 'h' to the height in equation 2 because the angle of elevation is measured from the bottom of the antenna.

3. Convert the degrees and minutes into decimal degrees for easier calculations.
51° 23' = 51 + 23/60 = 51.3833°
42° 10' = 42 + 10/60 = 42.1667°

4. Now we can substitute the values into equations 1 and 2:

tan(51.3833°) = h / 36.0 ft
tan(42.1667°) = (h + antenna height) / 36.0 ft

5. Rearrange equation 1 to solve for 'h':
h = tan(51.3833°) * 36.0 ft

6. Rearrange equation 2 to solve for the antenna height:
antenna height = (tan(42.1667°) * 36.0 ft) - h

7. Substitute the value of 'h' from equation 5 into equation 6 to find the antenna height.

8. Add the antenna height to 'h' to get the total height of the antenna.