Answer
50.6 km
Work Step by Step
We can construct a right angle triangle with the three vertices of the earth's center, the balloon, and the point on the horizon. The hypotenuse of the triangle is R+h, where R is the earth's radius and h is the height of the balloon. Let d be the distance from the balloon to the point on the horizon.
We can use the Pythagorean theorem to set up an equation and then solve it for d:
$R^2 + d^2 = (R+h)^2$
$R^2 + d^2 = R^2 + 2Rh + h^2$
$d^2 = 2Rh + h^2$
$d = \sqrt{2Rh + h^2} = \sqrt{(2)(6400)(0.2)+(0.2)^2} = 50.6~km$
