Question

A bacteria culture starts with 12,000 bacteria and the number doubles every 50 minutes.

(a) Find a formula for the number of bacteria at time t.


(b) Find the number of bacteria after one hour.


(c) After how many minutes will there be 50,000 bacteria?

Answer 1

a)
`y=12000(2)^{(t)/(50)}`

b) Approx. 27,569 bacteria

c) About 103 minutes

Explanation:

a)

This will follow exponential modelling with form of equation shown below:

`y=Ab^{(t)/(n)}`

Where

A is the initial amount (here, 12000)

b is the growth factor (double, so growth factor is "2")

n is the number of minutes in which it doubles, so n = 50

Substituting, we get our formula:

`y=Ab^{(t)/(n)}\\y=12000(2)^{(t)/(50)}`

b)

To get number of bacteria after 1 hour, we have to plug in the time into "t" of the formula we wrote earlier.

Remember, t is in minutes, so

1 hour = 60 minutes

t = 60

Substituting, we get:

`y=12000(2)^{(t)/(50)}\\y=12000(2)^{(60)/(50)}\\y=12000(2)^{(6)/(5)}\\y=27,568.76`

The number of bacteria after 1 hour would approximate be 27,569 bacteria

c)

To get TIME to go to 50,000 bacteria, we will substitute 50,000 into "y" of the equation and solve the equation using natural logarithms to get t. Shown below:

`y=12000(2)^{(t)/(50)}\\50,000=12,000(2)^{(t)/(50)}\\4.17=2^{(t)/(50)}\\Ln(4.17)=Ln(2^{(t)/(50)})\\Ln(4.17)=(t)/(50)*Ln(2)\\(t)/(50)=(Ln(4.17))/(Ln(2))\\(t)/(50)=2.06\\t=103`

After about 103 minutes, there will be 50,000 bacteria