Chapter 4, Problem 7T

(a)

To determine

To find: The model the populations after t hours.

Answer to Problem 7T

  $P\left(t\right)=1000{\left(8\right)}^t$

Explanation of Solution

Given: The initial size of a culture is 1000. After one hours the bacteria count is 8000.

The initial population, $P_0=1000$

After 1 hours, $P\left(1\right)=8000$

Let exponential function, $P\left(t\right)=P_0{\left(b\right)}^t$

  $\begin{array}{c}1000{\left(b\right)}^1=8000\\ b=8\end{array}$

Hence, the model is $P\left(t\right)=1000{\left(8\right)}^t$

(b)

To determine

To find: The population after 1.5 hours.

Answer to Problem 7T

22627

Explanation of Solution

Given: The model is $P\left(t\right)=1000{\left(8\right)}^t$

Put t=1.5 into model

  $\begin{array}{c}P\left(1.5\right)=1000{\left(8\right)}^{1.5}\\ \approx 22627\end{array}$

Hence, the population of bacteria after 1.5 hours approx. 22627.

(c)

To determine

To find: The time when population reach to 15,000

Answer to Problem 7T

1 hour 18 minutes

Explanation of Solution

Given: The model is $P\left(t\right)=1000{\left(8\right)}^t$

Put $P\left(t\right)=15,000$

  $\begin{array}{c}1000{\left(8\right)}^t=15000\\ 8^t=15\\ t=\frac{\ln 15}{\ln 8}\\ t\approx 1.3\end{array}$

Hence, after 1 hour 18 minutes populations reach 15,000.

(d)

To determine

To sketch: The graph of $P\left(t\right)=1000{\left(8\right)}^t$

Answer to Problem 7T

  

Explanation of Solution

Given: The model is $P\left(t\right)=1000{\left(8\right)}^t$

Graph the model using graphing calculator.