Expert: Paul Klarreich - 12/1/2007

Question
Hey, just needing some help with the grade 12 math homework. I have to questions to ask, if that's alright.

Q 1: A lighthouse is located at point A. A ship travels from point B to point C. The distance from the lighthouse to point B is 8.8km. The distance from point C to the lighthouse is 5.6km. Determine the distance from B to C, is <BAC is 120 degrees.

Q 2: A surveyor is measuring the height of a mountain. From point A, he measure the angle of inclination to the top of the mountain to be 51 degrees. From point B, 700m closer to the base of the mountain, the angle of inclination is 62 degrees. Determine the height of the mountain.


Thanks for your help!

Answer
Questioner:   Robyn
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometry
Question:  Hey, just needing some help with the grade 12 math homework. I have to questions to ask, if that's alright.

Q 1: A lighthouse is located at point A. A ship travels from point B to point C. The distance from the lighthouse to point B is 8.8km. The distance from point C to the lighthouse is 5.6km. Determine the distance from B to C, is <BAC is 120 degrees.

Q 2: A surveyor is measuring the height of a mountain. From point A, he measure the angle of inclination to the top of the mountain to be 51 degrees. From point B, 700m closer to the base of the mountain, the angle of inclination is 62 degrees. Determine the height of the mountain.


Thanks for your help!

============================
Hi, Rob,

1. Looks like a Law of Cosines, because you have two sides and the angle between them:

You have:

AB = c = 8.8
AC = b = 5.6
BC = a is to be found.
<BAC or <A = 120 degrees.

a^2 = b^2 + c^2 - 2bc cos A

a^2 = (5.6)^2 + (8.8)^2 - 2(5.6)(8.8) cos 120

I think you can do the rest.  It's calculator stuff.

..............................

2. Call the top of the mountain C, the 'base' of the mountain D, which you cannot see because there are all these rocks on top of it.

We need a plan.  Here it is:  First study triangle ABC, and use L. of Sines to find AB.  Then study right tri. ACD, and use the sine function to find CD.


In triangle ABC,
AB = c = 700
<A = 51
<B = <ABC = 118 degrees, the complement of the 'nearer' angle of elevation.
<C = <ACB = 11 degrees, by subtraction from 180

Now use Law of Sines to find AC = b:

 b         c
------ = -------
sin B    sin C

 b          700
-------- = -------
sin 118     sin 11


   700 sin 118
b = ------------
     sin 11


Use your calc:

b = 3239.1728376408833919239862244004 (Yes, it's big.  You thought this was a small mountain, perhaps.)

OK, now use the BIG triangle, ACD, which is a right triangle, with AC as the hypotenuse (we just found it).

CD = 3239.1728376408833919239862244004 sin 51.

CD = 2517.3100892337289585295724051391

That's your height.  It IS a small mountain.