Chapter 11.4, Problem 59E

To determine

The time to reach the population cell.

Answer to Problem 59E

  $90\min .$

Explanation of Solution

Given information:

Inspectors for the fulton county health department check food samples for the presence of the E. coli bacteria. When E. coli cells are placed in a medium that bprovides nuterients needed for growth, the bacteria population can increase exponentially, reproducing itself every $15$ minutes. If an inspector has a sample containing $1000$ bacteria cells, how long will it take the population to reach $64,000$ cells?

Calculation:

From the given information, sample containing $1000$ bacteria cells, and the rate of growth of the period is $64,000$ cells are as shown below:

The bacteria population increase exponentially by using $N=N_0{(1+r)}^t$ where $r$ is the rate of growth per time period, $t$ is the number of time periods, and $N$ is the final popualation number.

The bacteria reproduce itself every $15$ minutes so

  $r=100\%=1$

Find the value of $t$ :

  $\begin{array}{l}$ 64,000=1000{(1+1)}^t$$ 64,000=1000{(2)}^t$$ 64={(2)}^t$$ 2^6={(2)}^t({\text{because 64=2}}^6)$t=6\end{array}$

Now find how long will it take the population to reach $64,000$ cells:

The time to reach $64,000$ cells is:

  $t\times$ period $=6\times 15\min =90\min$

Thus, the time to reach cell is $90\min$ .