1. ## LOTR Question

Poor Hapless Bilbo is being carriend by a Giant Eagle flying 200ft off the ground; he doesnt know how fast he's going. He fixes his eyes on the base of a tree, dead ahead, and notices that it is at an angle of 50deg depressed from the horizontal. 60 seconds later, he has to look 80deg down from the horizontal to see the base of the tree.

How fast is Bilbo flying?

2. Hello, theqwertykid!

Did you make a sketch?

Poor Hapless Bilbo is being carriend by a Giant Eagle flying 200ft off the ground.
He fixes his eyes on the base of a tree dead ahead, and notices that
it is at an angle of 50° depressed from the horizontal.
60 seconds later, he has to look 80°down from the horizontal to see the base of the tree.

How fast is Bilbo flying?
Code:
      A    x    B
* - - - - * - - - - E
| * 50°   \ 80°
|   *       \
|     *      \
|       *     \
200 |         *    \
|           *30°\
|             *  \
|               * \
|             50° *\
- - * - - - - - - - - - * - -
G                   T

Bilbo was at point $\displaystyle A$; the tree is at $\displaystyle T.$
. . $\displaystyle AG = 200,\;\angle EAT = \angle ATG = 50^o$

He flew $\displaystyle x$ feet to point $\displaystyle B\!:\;\;\angle EBT = \angle BTG = 80^o$
. . Hence: .$\displaystyle \angle BTA = 30^o,\;\;\angle ABT = 100^o$

In right triangle $\displaystyle AGT\!:\;\;\sin50^o \:=\:\frac{200}{AT} \quad\Rightarrow\quad AT \:=\:\frac{200}{\sin50^o}$

In $\displaystyle \Delta ATB\!:\;\;\frac{x}{\sin30^o} \:=\:\frac{AT}{\sin100^o}\quad\text{(Law of Sines)}$

. . Then: .$\displaystyle x \;=\;\frac{200\sin30^o}{\sin50^o\sin100^o} \;=\;132.5545301$

Bilbo flew 132.55 feet in one miniute: .$\displaystyle \text{about }2.21\text{ ft/sec} \approx\:1.5\text{ mph}$