Answer:
The expression is: [tex]P(t) = 693e^{0.3662t}[/tex]
The initial number of strands is 693
Step-by-step explanation:
A bacterial culture is known to grow at a rate proportional to the amount present.
This means that the bacterial model can be modeled by the following equation:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(0) is the initial population, and r is the growth rate.
After one hour, there are 1000 strands of bacteria
This means that [tex]P(1) = 1000[/tex], So
[tex]P(0)e^{r} = 1000[/tex]
[tex]P(0) = \frac{1000}{e^{r}}[/tex]
After four hours, 3000 strands.
This means that P(4) = 3000. So
[tex]P(0)e^{4r} = 3000[/tex]
Since [tex]P(0) = \frac{1000}{e^{r}}[/tex], we have that:
[tex]\frac{1000}{e^{r}}e^{4r} = 3000[/tex]
[tex]\frac{e^{4r}}{e^{r}} = \frac{3000}{1000}[/tex]
[tex]e^{4r-r} = 3[/tex]
[tex]e^{3r} = 3[/tex]
[tex]\ln{e^{3r}} = \ln{3}[/tex]
[tex]3r = \ln{3}[/tex]
[tex]r = \frac{\ln{3}}{3}[/tex]
[tex]r = 0.3662[/tex]
The initial population is:
[tex]P(0) = \frac{1000}{e^{0.3662}} = 693[/tex]
The expression is:
[tex]P(t) = 693e^{0.3662t}[/tex]
The equation becomes [tex]y=694(1.44)^t[/tex]. The initial number of strands is 694.
An exponential growth is in the form:
y = abˣ
where y, x are variables, a is the initial value of y and b is the multiplier.
Let y represent the number of strands after t hours.
After one hour, there are 1000 strands of bacteria:
1000 = ab¹
ab = 1000 (1)
After four hour, there are 3000 strands of bacteria:
3000 = ab⁴
ab⁴ = 3000 (2)
Dividing equation 2 by 1:
b³ = 3
b = 1.44
ab = 1000
a = 1000/b = 1000/1.44 = 694
The equation becomes [tex]y=694(1.44)^t[/tex]
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