What is the height of the antenna (the distance from point A to point B), given the following information?

- Deepa stands at a horizontal distance of 27 meters from the building.
- The angle of elevation from her eyes to the roof (point A) is 35 degrees.
- The angle of elevation from her eyes to the top of the antenna (point B) is 41 degrees.
- Her eyes are 1.69 meters from the ground.

Options:
a) 17 meters
b) 21 meters
c) 24 meters
d) 30 meters

Question

High School

Answer :

wazzuhasaeed17 wazzuhasaeed17 · Answer 1 · 2025-01-10T21:16:00-05:00

The height of the antenna is found by calculating the heights of points A and B using trigonometry is 21.28 meters. The correct option is b.

Here's how to calculate the height of the antenna (AB) using trigonometry:

1. Set up the Diagram:

Imagine a right triangle ABC, where:

AB is the height of the antenna (unknown)

AC is the horizontal distance between Deepa and the building (27 meters)

BC is the vertical distance between Deepa's eyes and the roof (unknown)

DE is the height of Deepa's eyes (1.69 meters)

We are given the angle of elevation from Deepa's eyes to point A (∠BAC = 35°) and the angle of elevation to point B (∠ABC = 41°).

2. Solve for BC (height from eyes to roof):

Tangent (tan) function relates the opposite side (BC) to the adjacent side (AC) for a right triangle.

tan(∠BAC) = BC / AC

tan(35°) = BC / 27 meters

3. Solve for BC:

BC = AC * tan(35°) = 27 meters * tan(35°) ≈ 19.59 meters

4. Solve for AB (height of antenna):

Now that you know BC (height from eyes to roof), subtract Deepa's eye height (DE) to find the height of the antenna (AB).

AB = BC + DE = 19.59 meters + 1.69 meters ≈ 21.28 meters

The correct option is b.