Exponential Growth - Calculus

Question

3. A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.

(a) Find an expression for the number of bacteria after hours.
(b) Find the rate of growth after 3 hours.
(c) When will the population reach 10,000?

 

p242, #3

Answer 1

Hello Mia!

 

You will want to start with this formula:

 

A = Pe^rt

Where,

A = The final amount of bacteria

P = The initial amount of bacteria

e = The a mathematical constant that is the base of the natural logarithm (found on your calculator)

r = growth or decay rate

t = time

 

So, for part (a) you will define A(t) by first finding the growth rate (r) given that you initially have 100 bacterial (P) and after 1 hour (t) you have a final population of 420 bacteria (A) as:

 

A = Pe^rt  

420 = 100e^r(1)

4.2 = e^r(1)

ln(4.2) = r

 

So, you can define A(t) as,

A(t) = 100e^(ln(4.2))(t)

 

Part (b) asks for the growth rate so you will use the rate (r) used for above and the amount of bacteria after 3 hours (A(t)):

 

A(3) = 100e^(ln(4.2)(3)

A(3) = (100)(4.2³)

 

dP/dt = r(A(t)) =  ln(4.2)(A(3))

 

And (c) gives you a final population of 10,0000 bacteria (A) and you can now plug that into your expression from part (a) to find the time it will take to reach that population:

 

10000 = 100e^(ln(4.2))(t)

100 = e^(ln(4.2))(t)

ln(100) = (t)ln(4.2)

t = ln(100)/ln(4.2)

 

I will let you plug in the numbers and practice it yourself, let me know if you still have questions and I hope this has helped!