Trig application

Question

You approach a hill on top of which there is a tall radio antenna. You know from your map that your horizontal distance from the bottom of the radio antenna is 600 feet. The angle of elevation to the bottom of the antenna is , and the angle of elevation to the top of the antenna is . You figure that the height of the hill is  feet, and the height of the antenna is feet. (Enter your answers rounded to the nearest foot.)

Answer 1

Draw a triangle, where the bottom is 600 ft, and the opposite side is the hill height + the antenna.

Let's look at the hill part first. The angle to the top of the hill is 10 degrees, let's say the hill height is x.

The base is 600 ft. So we have the opposite side (x) and the adjacent side (600). What trig function uses opposite and adjacent? Think SOH CAH TOA.

Tan.So tan05 = x / 600 Multiply both sides by 600, and put into calculator. You get 105.8, rounded to nearest foot is 106. So the hill is 106 feet high.

Now let's look at another triangle with the antenna included in the height.

So we have a triangle where the angle of elevation is 25 degrees (to the hill plus the antenna). The base is still 600. Let's say the height of the hill + antenna is = y.

So tan25 = y / 600 Multiply both sides by 600, put into calculator. You get 279.78, rounded to nearest foot is 280.

280 feet is the height of the hill PLUS the antenna. We already know the height of the hill. How do we find the height of the antenna?

Hill + antenna = 280 Hill = 106 antenna = ?

Great job! Any questions?

-Margaret

Answer 2

Without a picture, I have to assume that the observer is at the bottom of the hill and measuring the angles to the bottom and top of the antenna from there. As we know the angles and horizontal distance to the antenna, we can write

tan 10 = height of hill/horizontal distance, therefore, the height to the bottom of the antenna (also top of hill) is given by height of hill = 400 ft x tan 10° = 71 ft. Similarly, the height of the top of the antenna from the observer's viewpoint is: antenna top = 400 ft x tan 25 = 187 ft. The height of the antenna is 187 - 71 = 116 ft.

Answer 3

You approach a hill on top of which there is a tall radio antenna. You know from your map that your horizontal distance from the bottom of the radio antenna is 600 feet. The angle of elevation to the bottom of the antenna is 10^o, and the angle of elevation to the top of the antenna is 25^o. You figure that the height of the hill is H feet, and the height of the antenna is A feet. (Enter your answers rounded to the nearest foot.)

Just re-writing the question with all the information, now onto how to solve it:

you would construct triangles and use trig signs to solve this, you have 2 right triangles (the right angle is where the horizontal distance from you to the hill and the height of the hill meet.

we will start with the height of the hill (H):

Your angle Θ is 10^o with a adjacent side length of 600ft, we want the height of the hill which is the opposite side so we will use tan.

tan(10^o)=H/600, 600tan(10^o)=H=106ft

Now we do the same process for the height of the antenna but this time your angle Θ is 25^o. Another important note, this will get you the height of the antenna and the mountain so we need to subtract out the mountain height once we find this height

tan(25^o)=A/600, 600tan(25^o)=A=280ft ---> Antenna height=280-106=174ft