**Understanding Exponential Growth in Bacterial Cultures**The number of bacteria in a culture is given by the function:\[ n(t) = 995e^{0.2t} \]where $ t $ is measured in hours.### Questions:**(a) What is the exponential rate of growth of this bacterium population?**Your answer is: \_\_\_\_\_\_\_\_\_\_\_ %**(b) What is the initial population of the culture (at $ t = 0 $)?**Your answer is: \_\_\_\_\_\_\_\_\_\_\_**(c) How many bacteria will the culture contain at time $ t = 3 $?**Your answer is: \_\_\_\_\_\_\_\_\_\_\_### Explanation:- **The Function:** The given function ($ n(t) = 995e^{0.2t} $) represents exponential growth, where $ n(t) $ is the number of bacteria at time $ t $, $ 995 $ is the initial population, and $ 0.2 $ is the growth rate per hour.- **Exponential Rate of Growth:** To find the exponential rate of growth, look at the exponent $ 0.2t $. The coefficient $ 0.2 $ indicates the rate. To express this as a percentage, multiply by 100: \[ 0.2 \times 100 = 20\% \]- **Initial Population:** The initial population can be found by setting $ t = 0 $: \[ n(0) = 995e^{0.2 \times 0} = 995e^0 = 995 \times 1 = 995 \]- **Population at $ t = 3 $:** Substitute $ t = 3 $ into the function: \[ n(3) = 995e^{0.2 \times 3} = 995e^{0.6} \] Using the approximate value $ e^{0.6} \approx 1.822 $: \[ n(3) \approx 995 \times 1.822 \approx 1813.89 \] Therefore, the approximate number of bacteria is 1814.

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The number of bacteria in a culture is given by the function n(t) = = 995e0.2t %3D where t is measured in hours. (a) What is the exponential rate of growth of this bacterium population? Your answer is % (b) What is the initial population of the culture (at t=0)? Your answer is (c) How many bacteria will the culture contain at time t=3? Your answer is

The number of bacteria in a culture is given by the function n(t) = = 995e0.2t %3D where t is measured in hours. (a) What is the exponential rate of growth of this bacterium population? Your answer is % (b) What is the initial population of the culture (at t=0)? Your answer is (c) How many bacteria will the culture contain at time t=3? Your answer is

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**Understanding Exponential Growth in Bacterial Cultures**

The number of bacteria in a culture is given by the function:

\[ n(t) = 995e^{0.2t} \]

where $ t $ is measured in hours.

### Questions:

**(a) What is the exponential rate of growth of this bacterium population?**

Your answer is: \_\_\_\_\_\_\_\_\_\_\_ %

**(b) What is the initial population of the culture (at $ t = 0 $)?**

Your answer is: \_\_\_\_\_\_\_\_\_\_\_

**(c) How many bacteria will the culture contain at time $ t = 3 $?**

Your answer is: \_\_\_\_\_\_\_\_\_\_\_

### Explanation:

- **The Function:**
The given function ($ n(t) = 995e^{0.2t} $) represents exponential growth, where $ n(t) $ is the number of bacteria at time $ t $, $ 995 $ is the initial population, and $ 0.2 $ is the growth rate per hour.

- **Exponential Rate of Growth:**
To find the exponential rate of growth, look at the exponent $ 0.2t $. The coefficient $ 0.2 $ indicates the rate. To express this as a percentage, multiply by 100:
\[
0.2 \times 100 = 20\%
\]

- **Initial Population:**
The initial population can be found by setting $ t = 0 $:
\[
n(0) = 995e^{0.2 \times 0} = 995e^0 = 995 \times 1 = 995
\]

- **Population at $ t = 3 $:**
Substitute $ t = 3 $ into the function:
\[
n(3) = 995e^{0.2 \times 3} = 995e^{0.6}
\]
Using the approximate value $ e^{0.6} \approx 1.822 $:
\[
n(3) \approx 995 \times 1.822 \approx 1813.89
\]
Therefore, the approximate number of bacteria is 1814.

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