Thread: Finding length in 3D Trigonometry...

Question:

David walks along a straight road. At one point he notices a tower on a bearing of 053 degrees with an angle of elevation of 21 degrees. After walking 230m, the tower is on a bearing of 342 degrees, with an angle of elevation of 26 degrees. Find the height of the tower correct to the nearest meter.

I have attempted to solve this question on numerous occassions, without success however.

I believe I had done the question correctly, working out the missing angles and then solving the height using the Sine Rule in the horizontal plane & then the basic trigonmetry function of Sin in one of the vertical planes, but however I have gotten the incorrect answer I am fairly sure, after I worked out the distance of one side using two different applications of the Sine Rule, and getting different answers!

Correct Answer by the way is 84m.. any help would be appreciated!

Matt

Hello, Matt!

Is there a typo?
The given measurements seem to be conflicting . . .

David walks along a straight road.
At one point he notices a tower on a bearing of 053° with an angle of elevation of 21°.
After walking 230m, the tower is on a bearing of 342° with an angle of elevation of 26°.
Find the height of the tower correct to the nearest meter

I believe your approach is correct; the fault may lie with the problem itself.

Looking down at the ground, we have this layout:

Code:

      P               T       Q
      :               *       :
      :             *  *      :
      :           * 71° *     :
      :      b  *        *    :
      :       *         a *18°:
      :     *              *  :
      :53°*                 * :
      : * 37°            72° *:
      * - - - - - - - - - - - *
      A           230         B

David walks 230 m from A to B.

/PAT = 53° . . . Hence: /A = /TAB = 37°

/QBT = 18° . . . Hence: /B = /TBQ = 72°


From the Law of Sines: .b .= .230·sin72°/sin71° .≈ .231.347

Now look at the vertical right triangle formed by point A and the tower.

Code:

                              S
                              *
                          *   |
                      *       |
                  *           | h
              *               |
          * 21°               |
      * - - - - - - - - - - - *
      A        231.347        T

We have: .h .= .231.347(tan21°) .≈ .88.8 m



If we solve for a = BT, we get: .a .≈ .146.393

Then from the vertical right triangle formed by point B and the tower,
. . we get: .h .= .146.393(tan26°) .≈ .71.4 m


We get two different heights for the tower. .Why?

(1) There could be a typo in the problem

(2) The road is straight but not horizontal.
. . .If that's the case, good luck!

Thanks for the help - you are correct I believe - if you assume that the road is not horizontal but is straight... you can get the correct answer!

Thanks again,

Matt