Answer:
It takes 3.15 hours for the population to double in size
Step-by-step explanation:
Given:
N= 100e^kt
N = 300
t = 5
To Find:
The time required for the population to double in size
Solution:
Step 1: Finding the value of k
[tex]N= 100e^{kt}[/tex]-------------------(1)
ln(N) = ln(100) +kt
[tex]k = (\frac{1}{t})\cdot ln(\frac{N}{100})[/tex]
On substituting the given values
[tex]k = (\frac{1}{5})\cdot ln(\frac{300}{100})[/tex]
[tex]k = (\frac{1}{5})\cdot ln(3)[/tex]
[tex]k =(0.2)\cdot (1.098)[/tex]
k = 0.2197
Now eq(1) becomes
[tex]N= 100e^{(0.22)t}[/tex]-------------------(2)
Step 2: Finding t value
[tex]ln(N) = ln(100) \cdot (0.22)t[/tex]
[tex]t = (\frac{1}{0.22}) \cdot ln(\frac{N}{100})[/tex]
N= 200
[tex]t = (\frac{1}{0.22}) \cdot ln(\frac{200}{100})[/tex]
[tex]t = (\frac{1}{0.22}) \cdot ln(2)[/tex]
[tex]t = (4.54) \cdot(0.693)[/tex]
t = 3.15
The time required for the population to double in size is 3.15 hours.
Given data:
The linear model to represent the number of bacteria is,
[tex]N = 100e^{kt}[/tex] ........................................................(1)
Here, k is the constant and t is the time required for the population to double in size.
Linear model is a mathematical tool to determine the rate of decay of any substance with respect to time.
Take log on both sides of the linear model as,
[tex]ln(N) =ln( 100e^{kt})\\\\ln(N) =ln( 100)+kt \times ln(e)\\\\ln(N) =ln( 100)+kt\\\\k =\dfrac{1}{t} \times ln\dfrac{N}{100}[/tex]
If N=300 when t= 5. Then the equation is evaluated as,
[tex]k =\dfrac{1}{5} \times ln\dfrac{300}{100}\\\\k=0.2197[/tex]
For double size of population,
[tex]N = 2 \times 100 = 200[/tex]
Then solve by substituting all the values in equation (1) as,
[tex]N = 100e^{kt}[/tex]
[tex](2 \times 100) = 100 \times e^{0.2197 \times t}\\200 = 100\times e^{0.2197 \times t}\\\\ln(2)= ln e^{0.2197 \times t}\\\\t = \dfrac{1}{0.2197} \times ln(2)\\\\t = 3.15 \;\rm hours[/tex]
Thus, the time required for the population to double in size is 3.15 hours.
Learn more about the linear model here:
https://brainly.com/question/24197246
