Chapter 9.9, Problem 29PPS

(a)

To determine

A table to show the bacteria population for the first 4 hours

Answer to Problem 29PPS

Time(hours) t	Bacteria$b=12{\left(3\right)}^t$
0	12
1	36
2	108
3	324
4	972

Explanation of Solution

Given:

A scientiost estimates that a bacteria culture with an initial population of 12 will triple every hours.

Concept Used:

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Calculation:

In order to make a table of bacteria population

Time(hours) t	Bacteria$b=12{\left(3\right)}^t$
0	12
1	36
2	108
3	324
4	972

Thus, the table is above for the population of bacteria every hour

(b)

To determine

Which kind of model best represents the data

Answer to Problem 29PPS

The model is exponential

Explanation of Solution

Given: the table

Time(hours) t	Bacteria$b=12{\left(3\right)}^t$
0	12
1	36
2	108
3	324
4	972

Concept Used:

If first difference of y-values is equal, then model be linear

If second difference be equal, then the model be quadratic

If first and second differences are not equal then the model will be exponential

Calculation:

In order to the models we see that in the table there is a constant ratio of 3 and the best models is a exponential function of the form $b=a{c}^t$

(c)

To determine

The function that models the data

Answer to Problem 29PPS

The function is $b=12{\left(3\right)}^t$ .

Explanation of Solution

Calculation:

In order to find the function we have constant ratio is 3 so $c=3$ to find a use an ordered pair from the table $\left(0,12\right)$ shown below:

  $\begin{array}{c}b=a{c}^x\\ 12=a{\left(3\right)}^0\\ 12=a\\ \Rightarrow a=12\end{array}$

Thus, the function is $b=12{\left(3\right)}^t$ .

(d)

To determine

How many bacteria after 8 hours

Answer to Problem 29PPS

The number of bacteria after 8 hours is $b=78732$ .

Explanation of Solution

Calculation:

In order to find the number of bacteria after 8 hours shown below:

  $\begin{array}{c}b=12{\left(3\right)}^t\\ =12{\left(3\right)}^8\\ =12\times 6561\\ b=78732\end{array}$

Thus, the number of bacteria after 8 hours is $b=78732$ .