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Answer :
Certainly! Let's walk through the process to solve this problem step by step.
1. Understanding the Problem:
We need to create a function that represents the population of bacteria growing exponentially over time. Initially, the population is 3700, and it grows at a rate of 11% per day. We also need to find the growth rate per hour.
2. Exponential Growth Formula:
The general formula for exponential growth is:
[tex]\[
P(t) = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the rate of growth per time unit (t), and
- [tex]\( t \)[/tex] is the number of time units.
3. Initial Values:
- Initial population ([tex]\( P_0 \)[/tex]) = 3700
- Daily growth rate = 11%
4. Converting Daily Growth Rate to Hourly:
Since we want the function in terms of hours, we need to find the equivalent hourly rate. We do this by using the property of exponential growth over shorter intervals:
First, convert the daily growth rate into a factor:
[tex]\[
1 + \text{daily rate} = 1 + 0.11 = 1.11
\][/tex]
Then, calculate the equivalent hourly growth factor:
[tex]\[
\text{Hourly factor} = 1.11^{(1/24)}
\][/tex]
After performing the calculation, we find that the hourly growth factor is approximately 1.0044.
5. Writing the Function:
Now, we can write the population function as:
[tex]\[
f(t) = 3700 \times (1.0044)^t
\][/tex]
where [tex]\( t \)[/tex] is in hours.
6. Finding the Growth Rate per Hour:
- The hourly growth rate in percentage can be found by subtracting 1 from the hourly factor and converting it to a percentage:
[tex]\[
\text{Hourly rate percentage} = (1.0044 - 1) \times 100 = 0.44\%
\][/tex]
So, the function representing the population of bacteria after [tex]\( t \)[/tex] hours is [tex]\( f(t) = 3700 \times (1.0044)^t \)[/tex], and the hourly growth rate is approximately 0.44%.
1. Understanding the Problem:
We need to create a function that represents the population of bacteria growing exponentially over time. Initially, the population is 3700, and it grows at a rate of 11% per day. We also need to find the growth rate per hour.
2. Exponential Growth Formula:
The general formula for exponential growth is:
[tex]\[
P(t) = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the rate of growth per time unit (t), and
- [tex]\( t \)[/tex] is the number of time units.
3. Initial Values:
- Initial population ([tex]\( P_0 \)[/tex]) = 3700
- Daily growth rate = 11%
4. Converting Daily Growth Rate to Hourly:
Since we want the function in terms of hours, we need to find the equivalent hourly rate. We do this by using the property of exponential growth over shorter intervals:
First, convert the daily growth rate into a factor:
[tex]\[
1 + \text{daily rate} = 1 + 0.11 = 1.11
\][/tex]
Then, calculate the equivalent hourly growth factor:
[tex]\[
\text{Hourly factor} = 1.11^{(1/24)}
\][/tex]
After performing the calculation, we find that the hourly growth factor is approximately 1.0044.
5. Writing the Function:
Now, we can write the population function as:
[tex]\[
f(t) = 3700 \times (1.0044)^t
\][/tex]
where [tex]\( t \)[/tex] is in hours.
6. Finding the Growth Rate per Hour:
- The hourly growth rate in percentage can be found by subtracting 1 from the hourly factor and converting it to a percentage:
[tex]\[
\text{Hourly rate percentage} = (1.0044 - 1) \times 100 = 0.44\%
\][/tex]
So, the function representing the population of bacteria after [tex]\( t \)[/tex] hours is [tex]\( f(t) = 3700 \times (1.0044)^t \)[/tex], and the hourly growth rate is approximately 0.44%.
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