7.2.4Practice: Modeling: Geometric Sequences
2 answers:
Answer:
1. I chose the tennis ball.
What do you know?
I know that a tennis ball will rebound 58 inches when dropped onto a hard surface from a height of 100 inches. All of the measurements are taken from the bottom of the ball.
What do you want to find out?
I want to find out how high the tennis ball will bounce on the 10th bounce.
What kind of answer do you expect?
I expect to find an answer that fits into a geometric sequence, and has a constant common ration.
2. Assume that the ball rebounds the same percentage on each bounce. Using the initial drop height and the height after the first bounce, find the common ratio, r.
Note: Round r to three decimal places. Use this formula:
The initial drop height=100 inches
The height on the first bounce=58% or 0.58
r=the common ratio
r=0.58/1.00
r=0.58
3. State the general version of the recursive formula.
a(1)
a(n)=a(n-1)+d
a(height on the first bounce)
a(n)=a(number of terms-1)+common difference
4. Find the recursive formula for the height of your ball. Remember that the 1st term, a1, is the height of the ball on the first bounce.
a(1)=58
a(n)=a(n-1)+0.58
5. Fill out the following table for your ball's height after the first 3 bounces. Note: Let n = bounce number. The height on the 1st bounce, n = 1, is given.
Table is attached below.
6. Write the explicit formula for the geometric sequence of the height of the ball on the nth bounce. Use the formula an = a1 • rn – 1. Remember that the 1st term, a1, is the height of the ball on the first bounce.
an=58*0.58^n-1
7. Using the explicit formula, find the height of the ball on the 10th bounce.
a10=58*0.58^10-1
a10=58*0.58^9
0.58^9=0.00742...
Multiply 58 by 0.00742…=0.4308...
a10=0.43
The height of the ball on the 10th bounce is about 0.43 inches or 0.4308
8. What are some factors that could affect the ball's bounce? Why might a ball bounce higher or lower than the regulated height?
Factors that could affect the ball’s bounce could be the height from which the ball is dropped, the air pressure inside of the ball, the hardness of the surface, and the actual properties of the ball. A ball may bounce higher or lower than the regulated height because of the force of which it is being dropped.
9. For the sport you chose, why would it matter if a player used a ball that bounced higher than the regulation?
In tennis, it could give one player an advantage to make the ball bounce higher towards the opponent(s). Returning a high-bouncing ball could be relatively difficult for the players, so this could also make the match harder than it would otherwise be.
I hope this helps! :)
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