Chapter 12.2, Problem 41E

a.

To determine

To calculate:

Equation that will determine no. of bacteria present after $t$ hours.

Answer to Problem 41E

Equation that will determine no. of bacteria present after $t$ hours is $b_t=\left(2b_0\right){\left(2\right)}^{2r}$ .

Explanation of Solution

Given information:

For every half an hour it is doubled. That means thecoomon ratio of eh series is $2$ .

Total given hours are $t$ .

Calculation:

Total given hours are $t$ . Thus total number of time intervals to increase the bacterium is $n=2t$ . From the first term of the series is $a_1=b_0\left(2\right)$

Apply the formula for the $n^{th}$ term of a geometric series.

  $a_n=a_1{r}^{n-1}$

Total bacterium presented in the colony after $t$ hour is,

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ b_t=\left(2b_0\right){\left(2\right)}^{2r-1}\\ b_t=\left(2b_0\right){\left(2\right)}^{2r}\end{array}$

Thus, Equation that will determine no. of bacteria present after $t$ hours is $b_t=\left(2b_0\right){\left(2\right)}^{2r}$ .

b.

To determine

To calculate:

Number. of bacteria present after $5$ hours.

Answer to Problem 41E

Number. of bacteria present after $5$ hours is $30270$ .

Explanation of Solution

Given information:

For every half an hour it is doubled. That means thecoomon ratio of eh series is $2$ .

Total given hours are $t$ .

Calculation:

Total given hours are $5$ . Thus total number of time intervals to increase the bacterium is $n=2t$ . From the first term of the series is $a_1=b_0\left(2\right)$

Apply the formula for the $n^{th}$ term of a geometric series.

  $a_n=a_1{r}^{n-1}$

It is given that $b_0=30,t=5$

Total bacterium presented in the colony after $5$ hour is,

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ b_t=\left(b_0\right){\left(2\right)}^{2\left(5\right)}\\ b_t=30270\end{array}$

Thus, Number. of bacteria present after $5$ hours is $30270$ .

c.

To determine

To explain:

Assumptions are made in using formula found in part (a).

Answer to Problem 41E

Assumptions made are an adequate food, oxygen supply, appropriate surrounding temperature and room for growth.

Explanation of Solution

It is assumed that favorable conditions are mentioned for the growth of the bacteria, such as an adequate food and oxygen supply, appropriate surrounding temperature and adequate room for growth.