Question

The doubling time of a bacterial population is 20 minutes. After 100 minutes, the bacterial population was 80000. What was the initial population of bacteria? Preview Round your answer to the nearest whole bacterium. Using your rounded answer from above, find the size of the bacterial population after 4 hours. Preview Round your answer to the nearest whole bacterium.

Answer 1

a) Population after t minutes is given by

P(t) = P_oe^kt

where P_o is the initial population

Doubling time = 20 minutes

This implies

P_oe^k(20) = 2P_o

=> e^20k = 2

=> ln e^20k = ln 2

=> 20k = ln 2

=> k = (1 / 20) ln 2

=> k = 0.034657

Therefore,

P(t) = P_oe^0.034657t

Given, Population after 100 minutes = 80000

=> P_oe^{0.034657 * 100} = 80000

=> P_oe^3.4657 = 80000

=> P_o = 2500

Therefore,

Initial Population = 2500

P(t) = 2500e^0.034657t

b) 4 hours = 240 minutes

Substituting t = 240 minutes in P(t) we get

Population after 4 hours

= 2500e^{0.034657 * 240}

= 10239118