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? Show your work.

Answer 1

Let's assume the distance between the airplane and the first fire is x feet.
Therefore, the distance between the airplane and the second fire is also x feet.

Using trigonometry, we can find the height of the airplane above the ground when it is directly above each fire.

In the first case, the angle of depression is 43°, and the height of the airplane above the ground is 2500 ft.
Using the tangent function: tan(43°) = height/ x
height = x * tan(43°)

In the second case, the angle of depression is 20°, and the height of the airplane above the ground is 2500 ft.
Using the tangent function: tan(20°) = height/x
height = x * tan(20°)

Since the height above the ground for both fires is the same, we can set the two equations equal to each other:
x * tan(43°) = x * tan(20°)

Simplifying the equation:
tan(43°) = tan(20°)

Using a calculator, we find that both tangent values equal approximately 0.36397.

Substituting this value back into either equation:
x * 0.36397 = 2500

Dividing both sides by 0.36397:
x ≈ 6861.76

Therefore, the distance between the two fires is approximately 6862 feet to the nearest foot.