Two fire towers, A and B, are 20.3 km apart. From tower A, the bearing of tower B is 70̊. The ranger in each tower observes a fire and radios the bearing of the fire from the tower. The bearing from tower A is 25̊ and from tower B is 345̊. How far is the fire from each tower?

triangle A B F

North to AB = 70
North to AF = 25
so
angle FAB = 70-25 = 45
North to BF = 345 clockwise so 15 counterclockwise
so exterior angle AB to BF = 70 +15 = 85
so
angle ABF = 180 -85 = 95
that leaves angle AFB = 180 -45-95 = 40
then
sin 40/20.5 = sin 95/af = sin 45/bf

ohh thanks

To find the distance of the fire from each tower, we can use trigonometry and the given bearings.

Let's start with tower A.

From tower A, the bearing to tower B is 70̊. This means that if we draw a line from tower A to tower B, it would form an angle of 70̊ with the line from tower A pointing north.

Next, we are given the bearing from tower A to the fire as 25̊. This means that if we draw a line from tower A to the fire, it would form an angle of 25̊ with the line from tower A pointing north.

Now, we can use these angles to find the distance of the fire from tower A.

We can draw a right triangle, with the line from tower A to the fire as the hypotenuse, the north line as the adjacent side, and the line from tower A to tower B as the opposite side.

Since we have the adjacent side (the line from tower A to the fire) and the opposite side (the line from tower A to tower B), we can use the tangent function to find the length of the fire from tower A.

Using the formula tan(angle) = opposite / adjacent, we can plug in the values:

tan(25̊) = opposite / 20.3 km

Rearranging the formula to solve for the opposite side, we get:

opposite = tan(25̊) * 20.3 km

Calculating this, we find that the distance of the fire from tower A is approximately 9.12 km.

Now, let's move on to tower B.

From tower B, the bearing to tower A is 180̊ - 70̊ = 110̊. This means that if we draw a line from tower B to tower A, it would form an angle of 110̊ with the line from tower B pointing north.

We are also given the bearing from tower B to the fire as 345̊. This means that if we draw a line from tower B to the fire, it would form an angle of 345̊ with the line from tower B pointing north.

Again, we can use these angles to find the distance of the fire from tower B.

Using the same method as before, we can set up a right triangle with the line from tower B to the fire as the hypotenuse, the north line as the adjacent side, and the line from tower B to tower A as the opposite side.

Using the tangent function, we have:

tan(345̊) = opposite / 20.3 km

Solving for the opposite side, we get:

opposite = tan(345̊) * 20.3 km

Calculating this, we find that the distance of the fire from tower B is approximately 8.53 km.

Therefore, the fire is approximately 9.12 km from tower A and 8.53 km from tower B.