Please explain how to figure out the following 7th grade math problem: If a rubber ball bounces exactly 1/2 the height from which it is dropped; and it is dropped from the top of a building that is 64 meters tall, how high will the ball bounce on its eighth bounce?
Please explain how to figure out the following 7th grade math problem: If a rubber ball bounces exactly 1/2 the height from which it is dropped; and it is dropped from the top of a building that is 64 meters tall, how high will the ball bounce on its eighth bounce?
64(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + etc.
Get the idea?
In algebra terms, you could look at the problem like this:
A. Set your variables. Do this by thinking about which parts of the problem can change. For this problem, there is the height of the final bounce, the height of the building, and the number of bounces.
1) h = Height of final bounce
2) b = Height of building
3) n = Number of bounces
B. Look for the pattern (see tchrwill's example above).
Looks like in the fraction, the numerator "1" stays the same. The denominator changes though, as follows.
1 bounce = 1/2
2 bounces = 1/4
3 bounces = 1/8
4 bounces = 1/16
etc.
If you look closely, denominator seems to be powers of 2, such as 2exp(1) = 2; 2exp(2) = 4; 2exp(3) = 8; 2exp(4) = 16; and so on.
C. Try making a formula.
Based on the numbers above, the formula should probably be:
h = b x [1/2exp(n)]
Just think through this formula step by step and you'll see how I got that.
D. Test it (see if the formula works).
For the eighth bounce, we have
h = b x [1/2exp(n)]
h = 64m x [1/2exp(8)]
Note that 2exp(8) = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2. Multiply those out and you get 256.
So,
h = 64m x [1/256]
Note that 1/256 approximately equals 0.004. Get this by long division or by calculator.
Finally,
h = 64m x 0.004
h = 0.256m
So on the eighth bounce, the ball should bounce about 256mm.
You should also plug in n = 1 and n = 2 just to satisfy yourself that the final answer is right.
Hope that helps!
Doug Jobes
your formula is not right doug because there is already a formula which says b(common ratio exp(n-1)=h of n..
To figure out how high the ball will bounce on its eighth bounce, we first need to understand the pattern of the bounces.
Given that the ball bounces exactly 1/2 of the height from which it is dropped, we can create a pattern of the bounces:
First bounce: The height of the first bounce is 1/2 * 64 meters (since it's dropped from a 64-meter tall building).
Second bounce: The height of the second bounce is 1/2 * (1/2 * 64) meters (since the ball bounces at exactly 1/2 of the previous bounce height).
Third bounce: The height of the third bounce is 1/2 * (1/2 * (1/2 * 64)) meters.
And so on...
To find the height of the ball's eighth bounce, we need to extend this pattern to the eighth term. We can create an equation to represent the height of the ball's eighth bounce:
Height of eighth bounce = 1/2 * (1/2 * (1/2 * (1/2 * (1/2 * (1/2 * (1/2 * 64))))))
This represents eight layers of dividing the height by 2.
Simplifying the equation step by step:
1/2 * 64 = 32 (height of the first bounce)
1/2 * 32 = 16 (height of the second bounce)
1/2 * 16 = 8 (height of the third bounce)
1/2 * 8 = 4 (height of the fourth bounce)
1/2 * 4 = 2 (height of the fifth bounce)
1/2 * 2 = 1 (height of the sixth bounce)
1/2 * 1 = 0.5 (height of the seventh bounce)
1/2 * 0.5 = 0.25 (height of the eighth and final bounce)
Therefore, the ball will bounce to a height of 0.25 meters on its eighth bounce.