A plane is taking off at a 15° angle from a horizontal runway. When the altitude of the plane is 250 feet, apply trigonometric functions and the Pythagorean Theorem to find x, the horizontal distance the plane has flown.
To find x, we need to first determine the height of the plane above the horizontal runway when it has flown a horizontal distance of x.
Let h be the height of the plane above the horizontal runway when it has flown a horizontal distance of x. Given that the angle of takeoff from the horizontal runway is 15 degrees, we can determine the height h as follows:
h = 250 feet + x * tan(15°)
Now we can use the Pythagorean theorem to find the horizontal distance x:
x^2 + h^2 = (250 + x * tan(15°))^2
x^2 + (250 + x * tan(15°))^2 = 250^2
Expand the terms and simplify:
x^2 + 62500 + 500x*tan(15°) + x^2 * tan^2(15°) = 62500
Simplify further:
2x^2 + 500x*tan(15°) + x^2 * tan^2(15°) = 0
Rearranging the equation:
x^2 + 500*x*tan(15°) + x^2*tan^2(15°) = 0
We can now solve this quadratic equation to find the horizontal distance x the plane has flown.