An airplane is flying directly toward two forest fires. From the airplane, the angle of depression to one fire is 43° and 20° to the other fire. The airplane is flying at an altitude of 2500 ft. What is the distance between the two fires to the nearest foot?

Answer 1

Let's assume that the distance between the airplane and the first fire is x feet.
In this case, the distance between the airplane and the second fire would be (x + d) feet, where d is the distance between the two fires.
From the information given, we can set up the following equations:
tan(43°) = 2500 / x
tan(20°) = 2500 / (x + d)
To find the distance between the two fires, we need to solve these equations for x and d.
From the first equation, we can isolate x by multiplying both sides by x:
x * tan(43°) = 2500
x = 2500 / tan(43°)
Using a calculator, we find that x is approximately 2500 / tan(43°) ≈ 2500 / 0.932 = 2677.93 feet.
Now we can substitute this value into the second equation to solve for d:
tan(20°) = 2500 / (2677.93 + d)
tan(20°) * (2677.93 + d) = 2500
2677.93 + d ≈ 2500 / tan(20°)
2677.93 + d ≈ 2500 / 0.3639
2677.93 + d ≈ 6874.74
d ≈ 6874.74 - 2677.93
d ≈ 4196.81 feet.
Therefore, the distance between the two fires is approximately 4196.81 feet, to the nearest foot. Answer: \boxed{4197}.