Chapter 7.7, Problem 32PPE

(a)

To determine

Forming table, graph and equation.

Explanation of Solution

Given data:

Starting with 128 teams half of the remaining team eliminated each round.

Formula used:

An exponential decay function is

  $y=a\cdot b^x$

a is initial value

b is growth rate

Calculation:

Initial number of team is 128

As every time half team eliminated so growth factor is $b=\frac{1}{2}$

So the function is $y=128{(\frac{1}{2})}^x$

x	1	2	3	4	5	6	7
y	64	32	16	8	4	2	1

  

Conclusion:

Its an exponential graph.

(b)

To determine

Is it possible to remain 24 teams after a round.

Answer to Problem 32PPE

No it is not possible

Explanation of Solution

Given data:

  $y=128{(\frac{1}{2})}^x$

Calculation:

No it is not possible as by the graph in part (a) it can be seen that there is no value which is equal to 24 on the integer.So graph is the easiest way to represent the situation.

Conclusion:

No it is not possible

(c)

To determine

Domain of the function.

Answer to Problem 32PPE

The domain of the function is $0\le x\le 7$

Explanation of Solution

Given data:

  $y=128{(\frac{1}{2})}^x$

Calculation:

By the table in part (a) the starting value was 0 as first round than second round started which is x=1 and than third round is x=2 and at x=7 there is only 1 team left

So the domain of the function is $0\le x\le 7$

The domain of function represent the number of team left after each round.

Conclusion:

The domain of the function is $0\le x\le 7$.

(d)

To determine

Number of team left after 5 rounds.

Answer to Problem 32PPE

4 teams were left after 5 rounds.

Explanation of Solution

Given data:

5 rounds were completed.

Formula used:

  $y=128{(\frac{1}{2})}^x$

Calculation:

Number of teams after 5 rounds:

  $y=128{(\frac{1}{2})}^x$

  $x=5$

  $y=128{(\frac{1}{2})}^5$

  $y=\frac{128}{32}$

  $y=4\text{ teams}$

Conclusion:

4 teams were left after 5 rounds.