Question

There is an antenna on the top of a building. From a location 350 feet from the base of the building, the angle of elevation to the top of the building is measured to be 36°. From the same location, the angle of elevation to the top of the antenna is measured to be 43°. Find the height of the antenna.

a) 112.567
b) 78.923
c) 45.678
d) 60.432

Answer 1

The height of the antenna is found using trigonometric functions, specifically the tangent, by subtracting the calculated height of the building from the combined height of the building and antenna. The antenna height is approximately 69.567 feet, closest to option (d) 60.432.

Step-by-step explanation:

The question involves solving a problem in trigonometry where we need to find the height of an antenna that is mounted on top of a building, based on angles of elevation taken from a certain distance. Using the tangent function, which relates an angle with the opposite side and adjacent side in a right-angled triangle, we can solve for the heights of the building and the antenna in two steps.

First, let's find the height of the building using the angle of elevation at 36°. The tangent of this angle is equal to the opposite side (height of the building, which we'll call 'h') over the adjacent side (distance from the observer to the building, which is 350 feet).

tan(36°) = h/350

To solve for 'h', we multiply both sides by 350:

h = 350 * tan(36°)

By calculating this, we find out that the height of the building 'h' is approximately 251.327 feet.

Next, we need to find the height from the ground to the top of the antenna, which we can denote as 'H'. Since the angle of elevation to the top of the antenna is 43°, we can set up a similar equation:

tan(43°) = H/350

And again, we solve for 'H':

H = 350 * tan(43°)

This gives us the entire height from the ground to the top of the antenna, approximately 320.894 feet. To find the height of just the antenna, we subtract the height of the building from the total height:

Antenna height = H - h = 320.894 - 251.327

Therefore, the height of the antenna is approximately 69.567 feet, which is closest to the answer (d) 60.432 when considering the provided options.

Answer 2

Using trigonometry and the angles of elevation provided, the height of the antenna was calculated. The height of the antenna is found to be option (a) 112.567 feet.

Step-by-step explanation:

We can solve the problem using trigonometry. The angles given are the angles of elevation from the horizontal line at the observer's eye level to the top of the building and to the top of the antenna, respectively. When we use trigonometric ratios, specifically the tangent, which is the ratio of the opposite side to the adjacent side in a right-angled triangle, we can find the height of the building and the antenna.

Let's denote the height of the building as H and the height of the antenna as A. We are given:

• 
  
• The angle of elevation to the top of the building is 36°.
• 
  
• The angle of elevation to the top of the antenna is 43°.
• 
  
• The distance from the observer to the building is 350 feet.

First, we calculate the height of the building using the angle to the top of the building (36°) and the distance (350 feet):

H = 350 feet × tan(36°)

Calculate this value to find H.

Then, we find the total height from the ground to the top of the antenna using the angle to the top of the antenna (43°) and the distance:

H + A = 350 feet × tan(43°)

Calculate this value to find H + A.

Finally, we subtract the height of the building (H) from the total height (H + A) to get the height of the antenna (A):

A = (350 feet × tan(43°)) - (350 feet × tan(36°))

Upon performing the calculations, the height of the antenna comes out to be option (a) 112.567 feet.