"Assume that the population of a culture of bacteria is increasing exponentially.

Question

 If the population is 200 at 14:00, and it becomes 3000 at 15:00, what is the population a t17:00?

Answer 1

The population increases from 200 to 3000 in 1 hour, i.e. by a factor of 3000/200=15.

 

Since the growth is exponential, in 3 hours it will increase by a factor of 15³=3,375, to 200*3,375=675,000.

Answer 2

A = P*b^t

 

Let t be the time since 14:00.

 

200 = P*b^0 = P*1 = P

So A = 200*b^t

 

3000 = 200*b^(15-14) = 200*b^1

b = 3000/200 = 15

 

At 17:00:

A = 200*15^3 = 200*5*3*15^2 = 3000*225 = 675,000.

Answer 3

You will want to use the equation:

A=A₀e^rt

with A=the population after a certain time, t (in hours)

and A₀=the initial population at time t=0.

 

Using the two times that were given, you need to find r, the constant of exponential increase.

 

3000=200e^r(1)

15=e^r

ln(15)=ln(e^r)

 

According to logarithm expansion rules:

 

ln(15)=r(ln(e))

 

However, ln(e)=1, so:

 

ln(15)=r

 

Next, use this r in your calculations for the population at t=17, two hours later.

 

A=3000e^(ln(15))*(2)

 

According to logarithm operations:

 

A=3000e^ln(225)

A=3000(225)

A=675000 bacteria

 

If you would like more explanation about the logarithmic rules I used, please ask or arrange a lesson. I did not include them in the interest of focus and brevity.