Aball is dropped from a height of 10 feet and returns to hide that is one half the height from which it fell. debok continues to balance the half of the height of the previous balance each time. how far will the ball have traveled when it hits the ground for the fourth time

We can solve this problem by determining how far the ball travels on each bounce.  Once it has hit the ground the 4th time, it will have gone:

Down, up, down, up, down, up, down.

Now, we calculate how far each of these distances are.

Since the first drop is 10 feet, the ball will go up to 5 feet.  Then it will fall 5 feet and go up 2.5 feet.  The ball will then fall 2.5 feet, and rise 1.25 feet and fall 1.25 feet.  Now, we add all these numbers up.

10 + 5 + 5 + 2.5 + 2.5 + 1.25 + 1.25 = 27.5, so the ball will have travelled 27.5 feet.