Marna is flying a helicopter at 1,350 feet and sees an airplane 2,400 feet in front of but also above her. Marna knows the angle of elevation is 54 degrees. How far is the airplane from the ground?
To find the distance of the airplane from the ground, we first need to find the vertical distance between Marna's helicopter and the airplane.
The vertical distance can be calculated using the sine function:
sin(54 degrees) = vertical distance / 2400 feet
vertical distance = 2400 feet * sin(54 degrees)
vertical distance = 2400 feet * 0.809 = 1941.6 feet
Now, to find the distance of the airplane from the ground, we need to find the horizontal distance between Marna's helicopter and the airplane. This can be calculated using the cosine function:
cos(54 degrees) = horizontal distance / 2400 feet
horizontal distance = 2400 feet * cos(54 degrees)
horizontal distance = 2400 feet * 0.5878 = 1410.7 feet
Now, we can use the Pythagorean theorem to find the distance of the airplane from the ground:
distance = sqrt((horizontal distance)^2 + (vertical distance)^2)
distance = sqrt((1410.7 feet)^2 + (1941.6 feet)^2)
distance = sqrt(1990449.29 + 3768882.56)
distance = sqrt(5759331.85)
distance ≈ 2399.95 feet
Therefore, the airplane is approximately 2400 feet from the ground.