Trigonometry

Aminekhadir · #1 $Old$ December 9th, 2008, 12:33 AM

This is a question involving Non-right angled triangle trigonometry...
I hope you can help me solving this question.
Thanks,
Amine

A boat sailing from A to B travels in a straight line until the captain realises he is off course. The boat is turned through an angle of 60 degrees, and then travels another 10 km to B. The trip would have been 4 km shorter if the boat had gone straight from A to B. How far did the boat travel?

Grandad · #2 $Old$ December 9th, 2008, 03:14 AM

Hi-

Quote:

A boat sailing from A to B travels in a straight line until the captain realises he is off course. The boat is turned through an angle of 60 degrees, and then travels another 10 km to B. The trip would have been 4 km shorter if the boat had gone straight from A to B. How far did the boat travel?

Suppose that the boat has sailed x km to a point C before the captain changes course. So you can draw a triangle ABC with the following information:

AC = x km
CB = 10 km
Angle ACB = 180 - 60 = 120 degrees
AB is 4 km less than (x + 10). So?

You can now use the Cosine Rule on this triangle: $AB^2=AC^2 + BC^2 -2.AC.BC.cos C$

Now remove brackets and simplify. (Be careful with the cosine of 120 degrees - it's negative.) If you do it right, it turns out to be a very easy equation for x.

If yours doesn't work out to be easy, let me know, and I'll give you a bit more help.

Grandad

Aminekhadir · #3 $Old$ December 9th, 2008, 09:06 AM

Hi, thanks for the help!
My equation turned out to be:

AB^2 = X^2 + 10^2 - 2 * X * 10 * cos(120)
Is this correct, or have I already done something wrong?

Soroban · #4 $Old$ December 9th, 2008, 09:45 AM

Hello, Aminekhadir!

Quote:

My equation turned out to be:
. . $AB^2 \:= \:x^2 + 10^2 - 2(x)(10)\cos120^o$

Is this correct? . . . . well yes, but . . .

Code:

                                  *
                          C   *
                          *  60°
                 x    * 120°*
                  *           * 10
              *                 *
          *                       *
      * - - - - - - - - - - - - - - *
      A           x + 6             B

Distance AB is 4 km less than $x + 10.$
. . Hence: . $AB \:=\:x+6$

Law of Cosines: . $(x+6)^2 \;=\;x^2 + 10^2 - 2(x)(10)\cos120^o$

This comes out to: . $x \:=\:32$

Don't forget to answer the question.

Aminekhadir · #5 $Old$ December 9th, 2008, 09:52 AM

Hi,
Thanks a lot for helping

,
but are you completely sure that the answer is 32? Isn't it 42? :S
Or am I doing something wrong again?

Grandad · #6 $Old$ December 10th, 2008, 02:19 AM

Hello Aminekhadir

Yes, the answer is 42, but x is 32, as Soroban said. Can you see why?

Grandad

Aminekhadir · #7 $Old$ December 28th, 2008, 05:43 AM

yes, I can see it now.
Thanks.