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.

Answer 1

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.