an airplane is flying at an altitude of 2500 feet parallel to the ground, and the ground distance to the start of the airport runway is 7500 feet.
What s the plane's angle of depression to the start of the runway?
What is the distance between the airplane and the airport runway?
An airplane is flying at an altitude of 2500 feet parallel to the ground, and the ground distance to the start of the airport runway is 7500 feet.
To find the plane's angle of depression to the start of the runway, you can use the tangent function.
The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the altitude of the airplane, which is 2500 feet, and the adjacent side is the ground distance to the start of the runway, which is 7500 feet.
Therefore, the tangent of the angle of depression can be found by dividing the opposite side by the adjacent side:
tangent(θ) = opposite/adjacent
tangent(θ) = 2500/7500
tangent(θ) = 1/3
To find the angle, you can use the inverse tangent function (also known as arctan or tan^-1):
θ = arctan(1/3)
Using a calculator, you can find that the angle of depression is approximately 18.43 degrees.
To find the distance between the airplane and the airport runway, you can use the Pythagorean theorem.
According to the Pythagorean theorem, for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the altitude of the airplane is the opposite side, which is 2500 feet, and the ground distance to the start of the runway is the adjacent side, which is 7500 feet. The distance between the airplane and the airport runway is the hypotenuse.
Therefore, using the Pythagorean theorem:
hypotenuse^2 = opposite^2 + adjacent^2
distance^2 = 2500^2 + 7500^2
distance^2 = 6,250,000 + 56,250,000
distance^2 = 62,500,000
Taking the square root of both sides:
distance = √62,500,000
Using a calculator, you can find that the distance between the airplane and the airport runway is approximately 7,905.7 feet.
To find the plane's angle of depression to the start of the runway, we can use trigonometry. Specifically, we can use the tangent function.
First, we can draw a right-angled triangle representing the situation. The altitude of the airplane (2500 feet) is the opposite side, and the ground distance to the start of the runway (7500 feet) is the adjacent side. The angle of depression we want to find is the angle between the line of sight from the airplane to the start of the runway and the horizontal ground.
Using the tangent function:
Tangent(angle) = Opposite side / Adjacent side
Tangent(angle) = 2500 / 7500
We can simplify this fraction:
Tangent(angle) = 1/3
To find the value of the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(1/3)
Using a calculator, we can find the angle to be approximately 18.43 degrees.
Now, to find the distance between the airplane and the airport runway, we can use the Pythagorean theorem. The hypotenuse of the right-angled triangle represents this distance.
Using the Pythagorean theorem:
Hypotenuse^2 = Opposite side^2 + Adjacent side^2
Hypotenuse^2 = 2500^2 + 7500^2
Hypotenuse^2 = 6250000 + 56250000
Hypotenuse^2 = 62500000
Taking the square root of both sides:
Hypotenuse = √62500000
Hypotenuse ≈ 7918.38 feet
Therefore, the plane is approximately 7918.38 feet away from the airport runway.