From a fire tower, A, a fire is spotted on a bearing of N42°E. From a second tower, B, the fire is on a bearing of N12°W. The two fire towers are 23 km apart, and A is N63°W of B. How far is the fire from each tower?
Just need help interpreting the diagram from the question. Thank you in advanced.
My diagram has triangle ABF, with F as the fire.
angle A = 75°, angle B = 51° and angle F = 180-75-51 = 54°
also AB = 23 km
so you use the sine law to find the other two sides,
first one: AF/sin51 = 23/sin54
AF = 23sin51/sin54 = 22.09 km
find BF in a similar way.
thanks so much!
Why A=75°
To interpret the diagram and solve this problem, let's break it down step by step:
1. The information states that there are two fire towers, A and B, and they are 23 km apart. You can draw two points on a piece of paper to represent tower A and tower B, and label them accordingly.
2. The information further specifies that tower A is N63°W of tower B. This means that if you draw a line connecting the two towers, the line's direction would be N63°W. To determine this, start from tower B and turn 63 degrees to the left (west) from the north direction.
3. Now, we are given the bearings from each tower to the fire. The fire is spotted on a bearing of N42°E from tower A and on a bearing of N12°W from tower B. These bearings represent the angles formed between the line connecting each tower to the fire and the north direction.
4. To find the distance of the fire from each tower, we can use trigonometry. Let's start with tower A. Since the fire is on a bearing of N42°E from tower A, the angle formed between tower A, the line connecting tower A to the fire, and the north direction is 42 degrees (N42°E - N = 42°). Since the two towers are 23 km apart, we can consider this as the base of a right-angled triangle. The question is asking for the distance of the fire from tower A, which represents the height of the triangle.
To find the height, we can use trigonometry's tangent function. Tangent is defined as the ratio of the opposite side (height) to the adjacent side (base) in a right triangle. Therefore, we have:
tan(42°) = height / 23 km
Rearranging the equation to solve for the height:
height = 23 km * tan(42°)
Using a calculator, compute tan(42°) and then multiply the result by 23 km to find the height.
5. Next, let's find the distance of the fire from tower B. Since the fire is on a bearing of N12°W from tower B, the angle formed between tower B, the line connecting tower B to the fire, and the north direction is 12 degrees (N12°W - N = 12°). Again, we can consider the distance between the two towers (23 km) as the base of a right-angled triangle, and we want to find the height.
Similarly, we can use the tangent function:
tan(12°) = height / 23 km
Rearranging the equation to solve for the height:
height = 23 km * tan(12°)
Calculate tan(12°) and then multiply the result by 23 km to find the height.
6. Once you have found the heights of the two triangles, those will represent the distances of the fire from each tower.
Remember to use a calculator for the trigonometric calculations.