Chapter 7, Problem 24SGR

(a)

To determine

To write: An exponential function that could be used to model the number of bacteria after $x$ hours if the number of bacteria changes at the same rate.

Answer to Problem 24SGR

An equation that models the number of bacteria is $y=5000{\left(1.240\right)}^x$ .

Explanation of Solution

Given:

The initial bacteria population is $5000$ and after 8 hours it was $28,000$ .

Calculation:

At the beginning of the experiment, the time is 0 hours and there are $5000$ bacteria cells. So, the y-intercept and the value of $a$ is $5000$ . When $x=8$ , the number of bacteria cells is $28,000$ .

Substitute these values into an exponent function to determine the value of $b$ .

  $\begin{array}{c}y=a{b}^x\\ 28000=5000b^8\\ \frac{28000}{5000}=b^8\\ 5.6=b^8\end{array}$

Simplify further.

  $\begin{array}{c}b={\left(5.6\right)}^{1/8}\\ \approx 1.240\end{array}$

Therefore, an equation that models the number of bacteria is $y=5000{\left(1.240\right)}^x$ .

(b)

To determine

To find: The number of bacteria that can be expected in the sample after 32 hours.

Answer to Problem 24SGR

There will be about $4,880,496$ bacteria cells after 32 hours.

Explanation of Solution

Given:

The initial bacteria population is $5000$ and after 8 hours it was $28,000$ .

Calculation:

Substitute 32 for $x$ in the equation $y=5000{\left(1.240\right)}^x$ .

  $\begin{array}{c}y=5000{\left(1.240\right)}^{32}\\ \approx 4,880,496\end{array}$

Therefore, there will be about $4,880,496$ bacteria cells after 32 hours.