Chapter 9.9, Problem 18PPS

(a)

To determine

To calculate: To estimate the number of bacteria after $10$ hours and graph the function.

Answer to Problem 18PPS

The number of bacteria after $10$ hours is $539$

Explanation of Solution

Given information: The population of bacteria in a culture increases according to the function $y=500{\left(2.1\right)}^{0.01t}$ , where $t$ represents the number of hours

Calculation:

Given that the population of bacteria in a culture increases according to the function $y=500{\left(2.1\right)}^{0.01t}$ , where $t$ represents the number of hours

Number of bacteria after $10$ hours is calculated by substituting $t=10$ in the above equation:

  $\begin{array}{l}y=500{\left(2.1\right)}^{0.01\times 10}\\ y=500{\left(2.1\right)}^{0.1}\\ y=538.505\\ y\approx 539\end{array}$

In order to plot the graph of the function, find the solutions of the equation

When $x=0$

  $\begin{array}{l}y=500{\left(2.1\right)}^{0.01\times 0}\\ y=500{\left(2.1\right)}^0\\ y=500\end{array}$

When $x=10$

  $\begin{array}{l}y=500{\left(2.1\right)}^{0.01\times 10}\\ y=500{\left(2.1\right)}^{0.1}\\ y=538.505\\ y\approx 539\end{array}$

When $x=100$

  $\begin{array}{l}y=500{\left(2.1\right)}^{0.01\times 100}\\ y=500{\left(2.1\right)}^1\\ y=1050\end{array}$

Plotting the points on graph:

  

Conclusion:

Hence, the number of bacteria after $10$ hours is $539$

(b)

To determine

To calculate: To identify the y-intercept and explain its meaning

Answer to Problem 18PPS

y-intercept is $500$ , which is equal to initial number of bacteria

Explanation of Solution

Given information: The population of bacteria in a culture increases according to the function $y=500{\left(2.1\right)}^{0.01t}$ , where $t$ represents the number of hours

Calculation:

Given that the population of bacteria in a culture increases according to the function $y=500{\left(2.1\right)}^{0.01t}$ , where $t$ represents the number of hours

y-intercept is calculated by substituting $t=0$ in the above equation:

  $\begin{array}{l}y=500{\left(2.1\right)}^{0.01\times 0}\\ y=500{\left(2.1\right)}^0\\ y=500\end{array}$

This means that initial number of bacteria are $500$

Conclusion:

Hence, y-intercept is $500$