This problem uses the Pythagorean Theorem.  The rope is tied to the stake in the ground and the apple balloon, in the completely vertical direction.  Assuming the rope is completely extended while the wind is blowing, this length of the rope represents the hypotenuse of a right angle triangle.  Let's call this side C.  When the wind picks up, and the balloon is now 6 feet away from the stake.  This represents a leg of this right angle triangle.  Let's call this side B.

The Pythagorean Theorem states that:

A^2 + B^2 = C^2

So all we need to do is find the other leg, or side A.

Plug in B = 6 ft, and C = 10ft.

A^2 + 6^2 = 10^2

A^2 + 36 = 100

A^2 = 64

A = 8 ft