Calculate the height Trigonometry

[Peter G.]

Hi

A surveyor is attempting to calculate the height of a Point P, on a building by taking measurements on a horizontal level ground. From a point A, the angle of elevation of P is 30 degrees. He then advances 20 m towards Point P, Point B, and measures an angle of elevation of 45 degrees.

Calculate the height of P above the ground: (Answer = 34m)

I tried several ways and I never get 34, only something around 27 m:

180-45 = 135
180-135-30 = 15 degrees.

20 / sin 15 = x / sin 30
= 38.63703305

Then:

sin 45 = O / H
sin 45 x 38.63703305 = O
O = 27.32

Where am I going wrong?

Thanks,
Peter G.

 

[DrummingAtom]

Try using tan and then make the distance on the ground 20 + x. Which would be tan (30) = h/(20+x). That will get you started.

 

[SammyS]

Your result looks good to me !

In fact, I got the same answer using the right triangle with the 30° angle.

 

[Peter G.]

Yes, I think maybe the book is wrong... Sometimes it happens!

But if you guys don't mind, how do I do with the right angle triangle?

 

[SammyS]

You used the 45 -- 45 right triangle, I used the 30 -- 60 right triangle, very similar to what you did.

 

[mege]

How do you know the hypotenuse given your information? (and as such, how can you use sin/cos since your hypotenuse changes with the change in location as well?)

This is a ratio of tangents as Drummingatom pointed out.

(I get ~34.64m when I quickly plug it into a calculator)

 

[Peter G.]

The hypothenuse I got using sine rule. I got the angle opposite to the 20 m by doing 180 - 30 - (180-45)

 

[mege]

Hmmm, I just wrote out the tanget solution and it comes up with 27.3205 (just like running it through sines does). I see my mistake in what I plugged into the calculator hastily, sorry (ironic that it came close to your book answer).

h = height, x = distance from 45* to building (thus 30* to building is x+20)

tan 45 = h/x = 1, x=h
tan 30 = x/(x+20)
... few steps later
(20*tan 30)/(1-tan 30) = x = 27.32