Question

A bacterial culture is known to grow at a rate proportional to the amount present. After one hour, there are 1000 strands of bacteria and after four hours, 3000 strands. Find an expression for the number of strands present in the culture at any time t. What is the initial number of strands

Answer 1

Answer:

The expression is: $P(t) = 693e^{0.3662t}$

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:

$P(t) = P(0)e^{rt}$

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 $P(1) = 1000$, So

$P(0)e^{r} = 1000$

$P(0) = \frac{1000}{e^{r}}$

After four hours, 3000 strands.

This means that P(4) = 3000. So

$P(0)e^{4r} = 3000$

Since $P(0) = \frac{1000}{e^{r}}$, we have that:

$\frac{1000}{e^{r}}e^{4r} = 3000$

$\frac{e^{4r}}{e^{r}} = \frac{3000}{1000}$

$e^{4r-r} = 3$

$e^{3r} = 3$

$\ln{e^{3r}} = \ln{3}$

$3r = \ln{3}$

$r = \frac{\ln{3}}{3}$

$r = 0.3662$

The initial population is:

$P(0) = \frac{1000}{e^{0.3662}} = 693$

The expression is:

$P(t) = 693e^{0.3662t}$

Answer 2

The equation becomes $y=694(1.44)^t$. 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 $y=694(1.44)^t$

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