From a helicopter 420m in the air, a pilot sees a forest fire directly to the east.
The angle of depression is 10°
A rangers tower is located directly to the west, at an angle of depression of 12°.
How far is the tower from the fire?
Code:
H
X - - - - - - - * - - - - - - - Y
12° / : \ 10°
/ : \
/ 78° : 80° \
/ : \
/ : \
/ :420 \
/ : \
* - - - - - - - + - - - - - - - *
T a A b F
The helicopter is at \(\displaystyle H.\;\) Its height is \(\displaystyle HA\,=\,420\) m.
The fire is at \(\displaystyle F.\;\;\angle YHF\,=\,10^o\;\;\Rightarrow\;\;\angle FHA\,=\,80^o\)
The tower is at \(\displaystyle T.\;\;\angle XHT\,=\,12^o\;\;\Rightarrow\;\;\angle THA\,=\,78^o\)
Let \(\displaystyle a\,=\,TA,\;\;b\,=\,AF\)
In right triangle \(\displaystyle HAT:\;\tan78^o\,=\,\frac{a}{420}\;\;\Rightarrow\;\;a\,=\,420\cdot\tan78^o\)
In right triangle \(\displaystyle HAF:\;\tan 80^o\,=\,\frac{b}{420}\;\;\Rightarrow\;\;b\,=\,420\cdot\tan80^o\)
Can you finish it now?