Fire lookout towers are used to locate fires so they can be put out as soon as possible. These towers often work in networks and if two towers can see a fire at the same time, they can determine the location of the fire vey accurately.
A.) Two forest rangers in towers see a fire. The angle at Tower A between tower B and the fire is measured to be 41.5 degrees. The angle at tower B between tower A and the fire is measured to be 87.3 degrees. If the towers are known to be 34.32 km apart, how far is the fire from each tower? (I know I need to use Sine, but I don't know how to complete the calculations)
You are right, you have to use the sine law
Since you know 2 angles, we can find the angle at F to be 51.2°
So AF/sin87.3 = 34.32/sin51.2
AF = 34.32sin87.3/sin51.2
calculator time .....
AF = 43.988...
repeat similar steps to find BF
To solve this problem, we can use the law of sines. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides.
Let's label the distance from Tower A to the fire as a, and the distance from Tower B to the fire as b.
Using the law of sines, we can set up the following equation:
sin(A) / a = sin(B) / b
In this case, A is the angle at Tower A between Tower B and the fire (41.5 degrees), and B is the angle at Tower B between Tower A and the fire (87.3 degrees).
Substituting the known values, the equation becomes:
sin(41.5) / a = sin(87.3) / b
To find the value of a, we rearrange the equation to isolate a:
a = (sin(41.5) / sin(87.3)) * b
Now, the distance between the towers (34.32 km) can be represented as the sum of the distances from each tower to the fire:
a + b = 34.32
Using the previous equation, we can substitute the expression for a:
(sin(41.5) / sin(87.3)) * b + b = 34.32
Now we can solve for b. Let's multiply through by sin(87.3) to eliminate the sine terms:
sin(41.5) * b + sin(87.3) * b = 34.32 * sin(87.3)
Combine like terms:
b (sin(41.5) + sin(87.3)) = 34.32 * sin(87.3)
Now we can solve for b:
b = (34.32 * sin(87.3)) / (sin(41.5) + sin(87.3))
Using a calculator, we can find:
b ≈ (34.32 * 0.999) / (0.655 + 0.999) ≈ 21.56 km
Finally, we can substitute the value of b back into the equation for a:
a = (sin(41.5) / sin(87.3)) * 21.56
Using a calculator, we can find:
a ≈ (0.656 / 0.999) * 21.56 ≈ 14.18 km
Therefore, the fire is approximately 14.18 km from Tower A and 21.56 km from Tower B.
To solve this problem, we can use the law of sines, which relates the ratios of sides in a triangle to the sines of its corresponding angles.
First, let's label the given information:
Angle at Tower A (opposite side AB): 41.5 degrees
Angle at Tower B (opposite side BA): 87.3 degrees
Distance between the towers: 34.32 km
Let's assume the distance from Tower A to the fire is x km, and the distance from Tower B to the fire is y km.
Now, we can apply the law of sines to find the lengths of the sides:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, we want to find the lengths of sides AB and BA, so we'll use the following ratios:
sin(A) / AB = sin(B) / BA
Rearranging the equation, we get:
AB / sin(A) = BA / sin(B)
Now, let's substitute the given values:
AB / sin(41.5) = BA / sin(87.3)
To find the lengths of the sides, we need to solve for AB and BA. Multiplying both sides of the equation by the corresponding sine value, we have:
AB = (sin(41.5) / sin(87.3)) * BA
Similarly, BA = (sin(87.3) / sin(41.5)) * AB
Now, we can substitute the given distance between the towers:
34.32 = (sin(41.5) / sin(87.3)) * BA
BA = (34.32 * sin(87.3)) / sin(41.5)
Similarly:
34.32 = (sin(87.3) / sin(41.5)) * AB
AB = (34.32 * sin(41.5)) / sin(87.3)
Using a calculator, we can evaluate these expressions to find the values of AB and BA.