How to set up?

[kwelsch]

A ranger in tower a spots a fire at a direction of 32 degrees. A ranger in fire tower B, which is 22 miles directly east of tower A, spots the same fire at a direction of 54 degrees. How far from tower B is the fire?

 

[Mrspi]

A ranger in tower a spots a fire at a direction of 32 degrees. A ranger in fire tower B, which is 22 miles directly east of tower A, spots the same fire at a direction of 54 degrees. How far from tower B is the fire?

Did you draw a diagram? I'd do that first. Let F be the location of the fire.

AB = 22 miles

angle BAF = 32

angle ABF = 54

Now, use the Law of Sines.

 

[soroban]

Hello, kwelsch!

Some clarification is required . . .

A ranger in tower \(\displaystyle A\) spots a fire at a direction of 32 degrees.
A ranger in fire tower \(\displaystyle B\), which is 22 miles directly east of tower \(\displaystyle A\),
. . spots the same fire at a direction of 54 degrees.
How far from tower \(\displaystyle B\) is the fire?

What is your definition of "direction"?
From where are those angles measured?


If those are bearings (measured CW from North), the diagram is wacky.

                *
      |       *               *
      |     *     |        *
      |32d*       | 54d *
      | *         |  *
  - - * - - - - - * - - - - -
      A    22     B

If the angles are measured counter-clockwise from East,
. . the problem can be solved.
But all that should have been clearly stated.

                              o F
                          *  *
                      * 22d *
                  *        *
              *           *
          * 32d     126d * 54d
      o - - - - - - - - o - - - - E
      A        22       B

We have: .\(\displaystyle \angle FAE = 32^o,\;\angle FBE = 54^o \quad\Rightarrow\quad \angle FBA = 126^o\)
. . Hence: .\(\displaystyle \angle F \,=\,22^o\)

Now use the Law of Sines to find side \(\displaystyle BF.\)