Chapter 9, Problem 26PT

(a)

To determine

The model that represents the data.

Answer to Problem 26PT

There is a common ratio of 2, so the best model would be an exponential function to fit the data

Explanation of Solution

Given information:

 CAR CLUB The table shows the number of car club members for four consecutive years after it began.

Time (years)	0	1	2	3	4
Members	10	20	40	80	160

Formula used:

Subtraction method is used.

Calculation:

First compare the first differences in the number of members:

  $\begin{array}{l}20-10=10\\ 40-20=20\\ 80-40=40\\ 160-80=80\end{array}$

Since they're not equal, compare the second differences:

  $\begin{array}{l}20-10=10\\ 40-20=20\\ 80-40=40\end{array}$

Since the second differences are not equal, look for a common ratio:

  $\begin{array}{l}\frac{20}{10}=2\\ \frac{40}{20}=2\\ \frac{80}{40}=2\\ \frac{160}{80}=2\end{array}$

There is a common ratio of 2, so the best model would be an exponential function to fit the data

Conclusion:

There is a common ratio of 2, so the best model would be an exponential function to fit the data

(b)

To determine

The function that models the data.

Answer to Problem 26PT

The function that best fits the data is:

  $y=10\left(2^x\right)$

Explanation of Solution

Given information:

 CAR CLUB The table shows the number of car club members for four consecutive years after it began.

Time (years)	0	1	2	3	4
Members	10	20	40	80	160

Formula used:

  $y=a\times 2^x$

Calculation:

A function that fits the data should be an exponential function with a ratio of 2. The function of the form $y=a\times 2^x$ . Use a point from the data given to determine the value of a.

  $\begin{array}{l}y=a\times 2^x\\ 10=a\times 2^0\\ a=10\end{array}$

The function that best fits the data is:

  $y=10\left(2^x\right)$

Conclusion:

The function that best fits the data is:

  $y=10\left(2^x\right)$

(c)

To determine

The car club members after 6 years.

Answer to Problem 26PT

There will be 640 members after 6 years.

Explanation of Solution

Given information:

 CAR CLUB The table shows the number of car club members for four consecutive years after it began.

Time (years)	0	1	2	3	4
Members	10	20	40	80	160

Formula used:

Substitution method is used.

Calculation:

To find the number of car club members after 6 years, use the function found in part b and plug in 6 for x.

  $\begin{array}{l}y=10\left(2^6\right)\\ y=10\left(64\right)\\ y=640\end{array}$

There will be 640 members after 6 years.

Conclusion:

There will be 640 members after 6 years.