Problem solving: calculate the growth of the bacteria in a petri dish and analyze the relation among variables.

Question

You are a medical technologist, and you need to know the growth of the bacteria. Based on your experiment the initial bacteria are 3200 bacteria and it is doubled every 40 minutes. To elaborate further you need to show all the details below:

1. Construct the model of growth of the bacteria. How many bacteria present after 6

hours and 9 hours?

1. How long would it take to have a population of 10,000 bacteria?

Answer 1

Bacteria grow at an exponential rate (1 bacterium doubles into 2, 2 double into 4, 4 double into 8, and so on). So we will need an exponential function to represent this model.

One way we can model this is using the equation:

P(t) = P₀*2^t/40

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After 6 hours (360 minutes) we get P(360) = 3200*2^360/40 = 1638400

After 9 hours (540 minutes) we get P(540) = 3200*2^540/40 = 37072760

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To reach a population of 10,000 bacteria:

10000 = 3200*2^t/40

Solving for t:

10000/3200 = 2^t/40

3.125 = 2^t/40 * Since t is in the exponent, to solve for t we will need to take the logarithm of both sides

log(3.125) = log(2^t/40) * Using properties of logarithms, the exponent can be moved to a coefficient

log(3.125) = t/40*log(2)

t = 40*log(3.125)/log(2)

t = 65.75 minutes

Answer 2

A(t) = 3200(2)^t/4

A(6) = 3200(2)^6/4

A(9) = 3200(2)^9/4

10000 = 3200(2)^t/4