High School

Cai is trying to find the height of a radio antenna on the roof of a local building. He stands at a horizontal distance of 21 meters from the building. The angle of elevation from his eyes to the roof (point A) is [tex]24^{\circ}[/tex], and the angle of elevation from his eyes to the top of the antenna (point B) is [tex]35^{\circ}[/tex]. If his eyes are 1.69 meters from the ground, find the height of the antenna (the distance from point A to point B). Round your answer to the nearest tenth of a meter if necessary.

Answer :

The height of the antenna is 0.54m.

Let the height of the antenna be represented by x, and the distance from Cai's eyes to the top of building A by y. And let z = x + y, so that:

Tan 24,

y = Tan 24 x 21

y = 0.4452 x 21

y = 9.3492

y = 0.93 m

Also, Tan 35,

z = Tan 35 x 21

z = 0.7002 x 21

z = 14.7042

z = 1.47 m

Therefore,

x = z - y

x = 1.47 - 0.93

x = 0.54

x = 0.54 m

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Final answer:

The height of the antenna is found using the tangent of the angles of elevation to points AA and BB and horizontal distance, leading to an approximate antenna height of 5.8 meters.

Explanation:

To find the height of the antenna, which is the distance from point AA to point BB, we can use trigonometry. Since we know the horizontal distance from Cai to the building and the angles of elevation to the roof and the top of the antenna, we can use the tangent function to find the heights from Cai's eye level to points AA and BB, and then subtract to get the height of the antenna itself.

First, we calculate the height to point AA:

Height to AA (from Cai's eyes) = horizontal distance * tan(angle to AA)

Height to AA (from Cai's eyes) = 21m * tan(24 degrees

Then, we calculate the height to point BB:

Height to BB (from Cai's eyes) = horizontal distance * tan(angle to BB)

Height to BB (from Cai's eyes) = 21m * tan(35 degrees)

We then subtract the height to AA from the height to BB to find the height of the antenna and add Cai's eye level height:

Height of antenna = Height to BB - Height to AA + Cai's eye level

Height of antenna = (21m * tan(35 degrees)) - (21m * tan(24 degrees)) + 1.69m

After performing the calculations and rounding to the nearest tenth of a meter:

Height of AA = 21m * tan(24 degrees) ≈ 9.1m (Cai's eye level is 1.69m high, so total height to AA is 9.1m + 1.69m)

Height of BB = 21m * tan(35 degrees) ≈ 14.9m (Including eye level: 14.9m + 1.69m)

The height of the antenna (BB-AA) = (14.9m + 1.69m) - (9.1m + 1.69m) ≈ 5.8m

The height of the antenna is approximately 5.8 meters.

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