a special rubber ball is dropped from the top of a wall that is 64 feet high. each time the ball bounces it rises half as high as the distance it fell. The ball is caught when it bounces 1foot high. how many times did the ball bounce?

1 bounce=32

2 bounce=16
3 bounce=8
4 bounce=4
5 bounce=2
6 bbounce=1

To find out how many times the ball bounces, we can follow these steps:

Step 1: Determine the initial height from which the ball is dropped. In this case, the ball is dropped from a wall that is 64 feet high.

Step 2: Calculate the distance the ball travels when it bounces. Each time the ball bounces, it rises half as high as the distance it fell. Therefore, the first bounce covers a distance of 64 feet.

Step 3: Determine the height the ball reaches after the first bounce. The ball rises half as high as the distance it fell, so after the first bounce, the ball reaches a height of 32 feet.

Step 4: Repeat steps 2 and 3 until the ball reaches a height of 1 foot. We will continue the calculation until the height reaches 1 foot or less.

Second bounce: Falls 32 feet and reaches a height of 16 feet.
Third bounce: Falls 16 feet and reaches a height of 8 feet.
Fourth bounce: Falls 8 feet and reaches a height of 4 feet.
Fifth bounce: Falls 4 feet and reaches a height of 2 feet.
Sixth bounce: Falls 2 feet and reaches a height of 1 foot.

Step 5: Count the number of times the ball bounced. In this case, the ball bounced a total of six times.

Therefore, the ball bounced six times before being caught when it bounces 1 foot high.

To determine the number of times the ball bounced, we can set up a mathematical sequence to represent the height of each bounce.

Let's start by noting that the ball initially drops from a height of 64 feet. After the first bounce, the ball rises to half the distance it fell, which is 64/2 = 32 feet.

Subsequent bounces will follow the same pattern: the ball falls from the last bounced height and then rises to half as high as the distance it fell.

We can express this sequence using the formula:

Bounce height(n) = (64/2^n)

where 'n' represents the number of bounces.

Now, we need to find the value of 'n' when the bounce height is 1 foot.

Setting up an equation:

1 = (64/2^n)

To solve for 'n', we can multiply both sides by 2^n:

2^n = 64

Since 2 raised to the power of 6 is 64, we can deduce that n = 6.

Therefore, the ball bounced 6 times before it was caught when it rose to a height of 1 foot.