Fire towers A and B are located 10 miles apart. They use the direction of the other tower as
0°. Rangers at fire tower A spots a fire at 42°, and rangers at fire tower B spot the same fire
at 64°. How far from tower A is the fire to the nearest tenth of a mile?
With solution please!! :'(
To solve this problem, we can use the law of sines. Let's label the triangle formed by the two fire towers and the location of the fire as ABC, with fire tower A at vertex A, fire tower B at vertex C, and the location of the fire at vertex B.
First, let's find angle C in triangle ABC. We know that the direction from fire tower A to fire tower B is 0°. The direction from fire tower A to the fire is 42°. Therefore, the angle at fire tower B (angle C) is 42° - 0° = 42°.
Next, let's find angle A in triangle ABC. We know that the direction from fire tower B to fire tower A is 180° (since the opposite direction is 0°). The direction from fire tower B to the fire is 64°. Therefore, the angle at fire tower A (angle A) is 180° - 64° = 116°.
Now, we have the two angles A and C. We can use the law of sines, which states:
sin(A) / a = sin(C) / c
where a is the side opposite angle A and c is the side opposite angle C.
In our case, side c is the distance between fire tower A and fire tower B, which is given as 10 miles.
Plugging in the values, we have:
sin(116°) / a = sin(42°) / 10
To find side a, we can rearrange the equation:
a = (sin(116°) / sin(42°)) * 10
Using a calculator, we find:
a ≈ 13.53 miles
Therefore, the fire is approximately 13.53 miles from fire tower A to the nearest tenth of a mile.
To solve this problem, we can use the concept of trigonometry and the properties of triangles. Let's break down the steps to find the distance from tower A to the fire.
Step 1: Understand the problem
We have two fire towers, A and B, located 10 miles apart. The direction of the other tower is considered as 0°. Rangers at tower A spot the fire at 42°, and rangers at tower B spot the fire at 64°. We need to find the distance from tower A to the fire.
Step 2: Visualize the problem
To better understand the problem, let's draw a diagram. Mark towers A and B, which are 10 miles apart from each other. Also, draw a directional line segment from each tower where the fire is spotted.
A --------------- B
/ |
/ |
/ |
Fire
Step 3: Use trigonometry ratios
Since we know the angles and the length of the base (distance between towers A and B), we can use trigonometry ratios to find the distance from tower A to the fire.
Step 4: Apply trigonometry
We can use the tangent function to find the distance.
The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the distance from tower A to the fire, and the adjacent side is the distance between towers A and B.
Mathematically, we have:
tan(42°) = distance from A to fire / 10 miles
To find the distance from A to the fire, we isolate the variable:
distance from A to fire = tan(42°) * 10 miles
Using a calculator, we find that tan(42°) ≈ 0.9004.
Therefore, the distance from tower A to the fire is approximately:
distance from A to fire = 0.9004 * 10 miles ≈ 9.004 miles
So, the distance from tower A to the fire, rounded to the nearest tenth of a mile, is approximately 9.0 miles.
see
http://www.jiskha.com/display.cgi?id=1330160622