A bacterial culture grows according to the logistic equation

N(t) = K / [1 + ((K - N₀)/N₀) e^{-rt}]

where N(t) is the cell number at time t (hours), N₀ is the initial cell number, K is the carrying capacity and r is the intrinsic growth rate (h⁻¹).

The culture is started with N₀ = 2 × 10³ cells, the carrying capacity is K = 1 × 10⁹ cells, and after t = 5 h the cell number is N = 5 × 10⁵ cells.

(i) Determine the intrinsic growth rate r (h⁻¹) to two decimal places.

(ii) Using this r, calculate the time required for the culture to reach half of the carrying capacity (N = K/2). Give your answer in hours to one decimal place.

Question

A bacterial culture grows according to the logistic equation

N(t) = K / [1 + ((K - N₀)/N₀) e^{-rt}]

where N(t) is the cell number at time t (hours), N₀ is the initial cell number, K is the carrying capacity and r is the intrinsic growth rate (h⁻¹).

The culture is started with N₀ = 2 × 10³ cells, the carrying capacity is K = 1 × 10⁹ cells, and after t = 5 h the cell number is N = 5 × 10⁵ cells.

(i) Determine the intrinsic growth rate r (h⁻¹) to two decimal places.

(ii) Using this r, calculate the time required for the culture to reach half of the carrying capacity (N = K/2). Give your answer in hours to one decimal place.

Accepted Answer

r = 1.10 h⁻¹; time to reach half of K = 11.9 h

Waec Model Questions Vol1 2025 Biology Question 8

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