I worked this problem out but I do not fully understand what I am doing.
Fire tower A is 30 kilometers due west of fire tower B.. A fire is spotted from the towers, and the bearings from A and B are N 76 degrees E and N 56 degrees W, respectively. Find the distance d of the fire from the line segment AB.
I got 14 kilometers.
If the fire is labeled F, then in ΔABF, the angles are
A = 14°
B = 34°
If the altitude from F to AB meets it at point P, then
h/AP = sin 14°
h/PB = sin 34°
so,
AP = h/sin14°
PB = h/sin34°
AP+PB=30
h/sin14° + h/sin34° = 30
h(1/sin14°+1/sin34°) = 30
h = 30/(1/sin14°+1/sin34°) = 5.07
I'd be interested in the steps you took to arrive at 14.
To find the distance of the fire from the line segment AB, we can use trigonometry and vector addition. Here's how you can break it down step by step:
1. Visualize the problem: Draw a diagram with fire tower A and B located on a Cartesian coordinate system. According to the problem, tower A is due west of tower B, which means tower A is to the left of B.
2. Determine the given bearings: A bearing represents an angle measured clockwise from the north direction. From tower A, the bearing is N 76 degrees E, which means the angle between the line segment AB and the north direction is 76 degrees measured to the east. From tower B, the bearing is N 56 degrees W, which means the angle between the line segment AB and the north direction is 56 degrees measured to the west.
3. Calculate the angle between AB and the north direction: Since the bearings are measured from the north direction to the line segment AB, we need to find the angle between the bearings to determine the direction of the line segment AB. Subtract the angle of bearing B from the angle of bearing A:
Angle of AB = angle of bearing A - angle of bearing B
= 76 degrees - 56 degrees
= 20 degrees
This means the line segment AB is oriented 20 degrees east from the north direction.
4. Decompose the vectors: Break down the vectors from towers A and B into their respective north and east components. Since fire tower A is positioned to the left (west) of fire tower B, we will consider it as a negative value.
- From tower A: The west direction is the negative of the east direction, so the east component is 0 and the north component is -30 km (negative because it is in the south direction).
- From tower B: The west direction is the negative of the east direction, so the east component is 0 and the north component is +30 km (positive because it is in the north direction).
5. Combine the vectors: Add the respective north and east components of the two vectors to find the coordinates of the fire in relation to the origin.
- North component: (-30 km) + (+30 km) = 0 km
- East component: 0 km + 0 km = 0 km
This means the fire is located exactly on the line segment AB.
6. Calculate the distance from AB: Since the fire is located on the line segment AB, the distance from the fire to AB is zero kilometers.
Therefore, the distance d of the fire from the line segment AB is 0 kilometers, not 14 kilometers. Double-check your calculations to make sure you didn't make an error.