Bacteria growth after a product substitution

It is known that a particular type of bacteria grows very well under certain controlled conditions in a specially prepared Petri dish. The researcher believes that the growth rate of such bacteria can be modelled by t2dxdt−2xt=−x2t^2\frac{dx}{dt}-2xt=-x^2, where xx milligrams is the amount of bacteria grown in the dish after tt hours.

(i) Using the substitution x=ut2x=ut^2, show that the differential equation can be reduced to dudt=−u2\frac{du}{dt}=-u^2. marks

(ii) Find xx in terms of tt, given that there was 0.2 milligrams of bacteria after 15 minutes. Hence find the amount of this particular type of bacteria after 4 hours. marks

(iii) Explain if this mathematical model is a realistic one. marks

[2012/NJC/Prelim/I/4]

Total marks: 8

Solution

(i)

Substitution

Using x=ut2x=ut^2, show that t2dxdt−2xt=−x2t^2\dfrac{dx}{dt}-2xt=-x^2 reduces to dudt=−u2\dfrac{du}{dt}=-u^2.

Hint

  1. Differentiate x=ut2x=ut^2 with the product rule.

Answer · (i)

u′=−u2u\prime=-u^2.

(ii)

Solving and using data

Find xx in terms of tt, given 0.2 mg after 15 minutes, and find the amount after 4 hours.

Explore this idea

Hint

  1. Separate −duu2=dt-\frac{du}{u^2}=dt.

  2. Convert minutes to hours: t=14t=\frac14.

  3. Back-substitute x=ut2x=ut^2.

Answer · (ii)

x=16t2/(16t+1)x=16t^2/(16t+1); x(4)=256/65≈3.94x(4)=256/65\approx3.94 mg.

t≥0t\ge0 in hours.

Explore the idea

The bacteria model

Each dash shows the slope the differential equation gives at that point. Change the starting value and the solution follows the dashes.

Starting value of x at t = 0.250.2
Read the solution at t4
Try an example
x at t = 4
3.938
Rate dx/dt there
1
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

The solution rises almost linearly: there is no upper limit.

Ideas to try
  1. Read the amount at t=4t=4 hours.
  2. Does the curve ever level off?
(iii)

Evaluating a model

Explain whether this model is realistic.

Hint

  1. Look at the long-term behaviour.

Answer · (iii)

Unlimited long-term growth is unrealistic in a finite dish.