determine height of tower from angles of elevation, distance

[vedmachka]

"Rapunzel is locked in a tall tower by the wicked queen. Her knight in shining armour is calling up to her "Rapunzel, Rapunzel, let down your hair...", thinking he can climb up her long braids and rescue her. He is facing north and can see Rapunzel's window at an angle of elevation of 20 degrees. The queen is observing the situation and is facing west. She can see the window at an angle of elevation of 18 degrees. The knight and the queen are 100m apart. Determine the height of Rapunzel's window above the ground to the nearest metre."

This is how far I got: the picture is in three dimensions. There are three right angle triangles. However, where I got stuck is that I only know the hypotenuse of one right angle triangle (100m.), one angle in another triangle and one angle in the third triangle. I am lost.

If anybody has any idea, it would be much appreciated....

Thanks a bunch!!!

 

[Denis]

Re: Can't seem to solve this problem...

Is Rapunzel waving a white hankie and yelling "SAVE ME, SAVE ME!!" :wink:
And after the Knight climbs up using her hair, HOW d'heck are both coming down after ?

K = Knight's position, Q = Queen's position, T = Top of tower, B = Bottom of tower

So your 3 right triangles are: KTB, QTB and KQB

Let t = TB, a = QB and b = KB

Triangle QTB: t = a TAN(18)
Triangle KTB: t = b TAN(20)
So b = a TAN(18) / TAN(20)
Let k = TAN(18) / TAN(20) ; then b = ak

Triangle KQB : sides are 100,a,ak ; so:
a^2 + a^2 k^2 = 100^2
a^2(1 + k^2) = 100^2
a = SQRT[100^2 / (1 + k^2)] ; ; since we know k, then:
a = 74.5992....

Since t = a TAN(18), then t = 24.2387....

Did this one in full because I was "interested" in it; don't expect full solution from me again!!

 

[soroban]

Re: Can't seem to solve this problem...

Hello, vedmachka!

"Rapunzel is locked in a tall tower by the wicked queen. Her knight in shining armour
is facing north and can see Rapunzel's window at an angle of elevation of 20 degrees.
The queen is facing west and can see the window at an angle of elevation of 18 degrees.
The knight and the queen are 100m apart.
Determine the height of Rapunzel's window above the ground to the nearest metre.

      Diagram 1                           Diagram 2           Diagram 3     R
                  R                 R                                       *
                  *                 *                                     *   *
               *  |                 |   *                            x  *       *  y
            *     | h             h |       *                         *           *
         * 20     |                 |        18 *                   *               *
      * - - - - - *                 * - - - - - - - - *           * - - - - - - - - - *
      K     x                               y         Q           K        100        Q

Diagram 1: the knight is at \(\displaystyle K\), Rapunzel is at \(\displaystyle R.\)
\(\displaystyle x\) = distance from K to the tower.
\(\displaystyle h\) = height of the tower.
Angle of elevation: 20 degrees

Diagram 2: the queen is at \(\displaystyle Q.\)
\(\displaystyle y\) = distance from Q to the tower.
Angle of elevation: 18 degrees

Diagram 3: looking down from above.
\(\displaystyle \angle R = 90^o\)


\(\displaystyle \text{Diagram 1: }\;\tan 20^o \:=\:\frac{h}{x} \quad\Rightarrow\quad h \:=\:x\tan20^o\;\;[1]\)

\(\displaystyle \text{Diagram 2: }\;\tan18^o \:=\:\frac{h}{y} \quad\Rightarrow\quad h \:=\:y\tan18^o\;\;[2]\)

. . \(\displaystyle \text{Equate [1] and [2]: }\;x\tan20^o \:=\:y\tan18^o \quad\Rightarrow\quad y \:=\:\frac{\tan20^o}{\tan18^o}x\;\;[3]\)


\(\displaystyle \text{Diagram 3: }\;x^2+y^2\:=\:100^2\;\;[4]\)


\(\displaystyle \text{Substitute [3] into [4]: }\;x^2 + \left(\frac{\tan20^o}{\tan18^o}x\right)^2 \;=\;100^2\)

. . \(\displaystyle \text{which simplifies to: }\;x^2\;=\;\frac{100^2\tan^2\!18^o}{\tan^2\!18^o + \tan^2\!20^o}\)

. . \(\displaystyle \text{Hence: }\;x \;=\;\frac{100\tan18^o}{\sqrt{\tan^2\!18^o + \tan^2\!20^o}}\)


\(\displaystyle \text{Now substitute into [1] . . .}\)

 

[vedmachka]

Re: Can't seem to solve this problem...

Thank you guys sooooooo much.... This has been extremely helpful!!! I can't believe it was this simple...