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
Algebra and Trigonometry (6th Edition)
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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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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01ced3bf-0a3b-4c7d-b69b-2526ca821194%2F959e3ac5-c257-4ce3-bb60-82b55fc85168%2F75jjwm1k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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