depression (distance of tower from fire)

[sarah.p]

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?

My answer is "163.4m" but I think it's wrong.

 

[tkhunny]

Re: depression

A rangers tower is located directly to the west

Unclear. West of what? Angle of depression from what?

Note: "Angle of Depression" and "Angle of Inclination" are ALWAYS measured from the horizontal.

 

[sarah.p]

west of the helicopter

and thats all it says..

 

[skeeter]

let x = horizontal distance from helo to the fire ...

tan(10) = 420/x
x = 420/tan(10)

let z = horizontal distance from the helo to the tower

tan(12) = 420/z
z = 420/tan(12)

distance between tower and fire = x + z

 

[soroban]

Hello, sarah.p!

Did you make a sketch?

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?

                        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?