Chapter 1.3, Problem 38E

(a)

To determine

To find: The number of bacteria present initially in a petri dish.

Answer to Problem 38E

The number of bacteria present initially in a petri dish was $100$ bacteria.

Explanation of Solution

Given information: The exponential number of bacteria in a petri dish culture after $t$ hours is $B=100e^{0.693t}$ .

Calculation:

At initial position the time will be 0. Substitute $0$ for $t$ in the exponential model.

  $\begin{array}{c}B=100e^{0.693t}\\ =100e^{0.693\times 0}\\ =100e^0\\ =100\end{array}$

Therefore, the number of bacteria present initially in petri dish was $100$ bacteria.

(b)

To determine

To find: The number of bacteria present in petri dish after $6$ hours.

Answer to Problem 38E

The number of bacteria present in petri dish after $6$ hours is $6394$ bacteria.

Explanation of Solution

Given information: The exponential number of bacteria in a petri dish culture after $t$ hours is $B=100e^{0.693t}$ .

Calculation:

The formula for the number of bacteria present is $B=100e^{0.693t}$ .

Substitute $6$ for $t$ in the formula.

  $\begin{array}{c}B=100e^{0.693t}\\ =100e^{0.693\times 6}\end{array}$

Use the calculator to find the value of $100e^{0.693\times 6}$ . Press “ON” button and enter the keystrokes as given below:

  $\boxed{100}\boxed{\times }\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{0.693}\boxed{\times }\boxed{6}\boxed{)}$

The value of $100e^{0.693\times 6}$ is $6394.35$ .

Therefore, the number of bacteria present in petri dish after $6$ hours is approximate $6394$ bacteria.

(c)

To determine

To find: The number of hours for the bacteria to reach $200$ and also the time taken to double the number of bacteria.

Answer to Problem 38E

The number of hours required for the bacteria to reach $200$ and also the time taken to double the number of bacteria is 1 hour.

Explanation of Solution

Given information: The exponential number of bacteria in a petri dish culture after $t$ hours is $B=100e^{0.693t}$ .

Calculation:

The formula for the number of bacteria present is $B=100e^{0.693t}$ .

Substitute $200$ for $B$ in the formula.

  $\begin{array}{c}B=100e^{0.693t}\\ 200=100e^{0.693t}\\ 2=e^{0.693t}\end{array}$

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation $e^{0.693t}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{0.693}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-5,5\right]$ by $\left[-5,5\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

So, the value of $\text{X}$ is $1.00021$ hour. The approximate value is $1$ .

The number of hours taken for the bacteria to grow from $100$ to $200$ is $1\text{h}$ . So, the time taken to double the number of bacteria is $1\text{h}$ .

Therefore, the number of hours required for the bacteria to reach $200$ and also the time taken to double the number of bacteria is 1 hour.