a television antenna is on the roof of the building. from a pt. on the ground 36.0 feet from the building , the angle of elevation of the top and the bottom of the antenna are 51 degree celsiusand 42 degree celsuis respectively. how tall is antenna?

The height of any object viewed from a horizontal distance d and angle of elevation θ is d*tan(θ).

Let's say the top of the roof viewed from the ground 50 ft away is 30°.
The height of the roof is therefore
H=50tan(30°)=28.9 feet.

Use the same formula to get the height from the ground of the top and bottom of the antenna. The height of the antenna is therefore the difference between the two heights from the ground.

The antenna is 12.05 ft tall.

To find the height of the antenna, we can use trigonometry. Let's first label the diagram to make it easier to understand:

G (Ground)
|\
| \
| \
| \
| \
| \
| \
| \
| T (Top of the antenna)
| /
| /
| /
| /
| /
| /
| /
|/
B (Bottom of the antenna)

Given:
Angle of elevation from G to T = 51 degrees
Angle of elevation from G to B = 42 degrees
Distance from G to the building (BT) = 36.0 feet

Let's find the length of the antenna (BT) first using trigonometry:
We know that tan(theta) = opposite / adjacent.

For the angle of elevation of the top (51 degrees):
tan(51) = BT / 36
BT = 36 * tan(51)

Similarly, for the angle of elevation of the bottom (42 degrees):
tan(42) = BT / 36
BT = 36 * tan(42)

Since BT is the same for both equations, we can equate them:
36 * tan(51) = 36 * tan(42)

Now, let's find the height of the antenna (TB):
TB = BT * tan(42)

Substituting the value of BT from the equation above, we get:
TB = (36 * tan(42)) * tan(42)

Evaluating this expression, we find that the height of the antenna is approximately 28.95 feet.

To find the height of the antenna, we can use trigonometry and the given angles of elevation. Here's how you can solve it:

Step 1: Draw a diagram of the situation. Label the base of the antenna "B", the top of the antenna "T", and the point on the ground below "P". Draw a right triangle with the vertical side as the height of the antenna (h), the horizontal side as the distance from the point on the ground to the building (36.0 ft), and the hypotenuse as the total distance from the point on the ground to the top of the antenna.

Step 2: Determine the lengths of the sides of the triangle using trigonometric functions. We'll start with the bottom angle of elevation of 42 degrees. We know that the tangent function (tan) is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the antenna (h) and the adjacent side is the distance from the point on the ground to the building (36.0 ft). So, we can write:

tan(42°) = h / 36.0 ft ....(Eq. 1)

Similarly, we'll use the angle of elevation of 51 degrees for the top of the antenna. Again, using the tangent function, we can write:

tan(51°) = (h + x) / 36.0 ft ....(Eq. 2)

where x is the horizontal distance from the top of the antenna to the building.

Step 3: Solve the two equations simultaneously to find the height of the antenna (h). Rearrange Eq. 1 to solve for h, which gives:

h = 36.0 ft * tan(42°)

Substitute this value of h in Eq. 2:

tan(51°) = (36.0 ft * tan(42°) + x) / 36.0 ft

Now, rearrange this equation to solve for x:

x = 36.0 ft * tan(51°) - 36.0 ft * tan(42°)

Step 4: Calculate the height of the antenna (h) by substituting the values obtained from steps 3 and 4 into Eq. 1:

h = 36.0 ft * tan(42°)

Now plug in the values of the angles and solve for h:

h = 36.0 ft * tan(42°)
h ≈ 32.995 ft

So, the height of the antenna is approximately 32.995 feet.