Answer:
To determine how far the fire is from tower B, we can use trigonometry and the concept of similar triangles.
1. First, let's draw a diagram to visualize the situation. Place tower A and tower B 15 miles apart, and mark the location of the fire. Connect tower A, the fire, and tower B to form a triangle.
2. Since the rangers at tower A and tower B both spot the fire, we have two angles: one at tower A (32°) and one at tower B (63°).
3. Notice that the two triangles formed (tower A to the fire and tower B to the fire) are similar. This means that their corresponding angles are equal.
4. Using the fact that the sum of angles in a triangle is 180°, we can find the third angle in each triangle:
Angle at the fire in triangle with tower A: 180° - 90° - 32° = 58°
Angle at the fire in triangle with tower B: 180° - 90° - 63° = 27°
5. Now, we have two angles and the included side (the distance between the towers) for both triangles. This allows us to set up a proportion to find the distance from tower B to the fire.
Let x be the distance from tower B to the fire.
Using the proportion: x / 15 = tan(27°), we can solve for x.
6. To find x, multiply both sides of the equation by 15:
x = 15 * tan(27°)
7. Using a calculator, evaluate tan(27°) and multiply by 15 to find the distance from tower B to the fire.
By following these steps and using trigonometry, we can determine the distance from tower B to the fire to the nearest mile.
Explanation: