The bacteria in a culture is increasing according to the law of exponential growth.

Question

The bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 after 4 hours. Find the initial population.

Answer 1

Let N_h represent the bacteria count at hour h.

We are given that N₂ = 125 and N₄ = 350. Also, we are told that N_h follows a geometric pattern. We are asked for the value of N₀.

Geometric growth patterns follow the rule that the ratio of neighboring population counts are constant, i.e.

N_h+1 / N_h = k, for some constant k.

We are given time in 2-hour blocks of time, so let us see what ratios of population counts look like in 2-hour blocks of time:

N_h+2 / N_h = (N_h+2 / N_h+!) ⋅ (N_h+1 / N_h) = k ⋅ k = k²

Thus, we can say that

k² = 350 / 125 = 2.8,

and so

N₂ / N₀ = 2.8

Solving for N₀,

N₀ = N₂ / 2.8 = 125 / 2.8 = 44.6429

Rounding to the nearest integer, we get N₀ = 45.