Question

Avani is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 21 meters from the building. The angle of elevation from her eyes to the roof ((point AA)) is 38^{\circ} ∘ , and the angle of elevation from her eyes to the top of the antenna ((point BB)) is 46^{\circ} ∘ . If her eyes are 1.66 meters from the ground, find the height of the antenna ((the distance from point AA to point BB)). Round your answer to the nearest tenth of a meter if necessary.

Answer 1

Explanation:

the scenario creates 2 right-angled triangles.

both have the same first leg : the horizontal distance from Avani's eyes to the building (21 m).

and both have a right angle (90°) at the point, where the horizontal distance meets the building.

the difference is now the second leg : the height of the building (starting at 1.66 m above ground), and the height of the building plus the height of the antenna (again starting at 1.66 m above ground).

another difference is the length of the line of sight (from Avani to AA, and from Avani to BB).

driving these differences is the difference in the angle at Avani (38° vs. 46°).

now, remember the law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

a, b, c are the sides of the triangle, A, B, C are the corresponding opposite angles of the triangle.

and remember : the sum of all angles in a triangle is always 180°.

what is the plan ?

we need to calculate the second leg of the larger triangle, and then the second leg of the smaller triangle and subtract that from the second leg of the larger triangle.

in other words :

(building + antenna) - building = antenna

so, we start with the larger triangle (up to BB).

the angle at Avani is 46°.

the angle at the building is 90°.

the angle at BB is then

180 - 90 - 46 = 44°.

21/sin(44) = (building + antenna)/sin(46)

(building + antenna) = 21×sin(46)/sin(44) =

= 21.74613659... m

now, for the smaller triangle (up to AA).

the angle at Avani is 38°.

the angle at the building is 90°.

the angle at AA is then

180 - 90 - 38 = 52°.

21/sin(52) = building/sin(38)

building = 21×sin(38)/sin(52) = 16.40699816... m

the height of the antenna is then again

(building + antenna) - building = 5.339138433... m

≈ 5.3 m

Answer 2

Let's call the height of the antenna "h".

First, we can use the angle of elevation of 38^{\circ} ∘ to find the height of point A above the ground.

tan(38^{\circ}) = \frac{h}{21}

h = 21 \cdot tan(38^{\circ})

h \approx 15.6

So point A is approximately 15.6 meters above the ground.

Next, we can use the angle of elevation of 46^{\circ} ∘ to find the height of point B above the ground.

tan(46^{\circ}) = \frac{h}{d}

h = d \cdot tan(46^{\circ})

We can find the value of "d" using the Pythagorean theorem.

d^2 = 21^2 + 15.6^2

d \approx 25.7

So the distance from point A to point B is approximately 25.7 meters.

Finally, we can use the height of point A and the distance from point A to point B to find the height of point B (the height of the antenna).

h = d \cdot tan(46^{\circ})

h \approx 25.7 \cdot tan(46^{\circ})

h \approx 23.2

Therefore, the height of the antenna is approximately 23.2 meters.