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: .$\displaystyle \text{about }2.21\text{ ft/sec} \approx\:1.5\text{ mph}$