The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours, there are 50 bacteria, and after 5 hours, there are 400 bacteria.

Question

The number of bacteria in the culture is increasing according to the law of exponential growth. After 2 hours, there are 50 bacteria, and after 5 hours, there are 400 bacteria.How many bacteria will there be after 6 hours? ___bacteria

Answer 1

Exponential growth can be modeled as:

B(t) = B₀(1 + r)^t where t, in this case is time in hours, B₀ is the number of bacteria at time t = 0, B(t) is the number of bacteria at any time t, and "r" is the growth rate.

Using the data point (2 hours, 50 bacteria) we can say:

50 = B₀(1 + r)² or B₀ = 50/(1 + r)²

Using the data point (5 hours, 400 bacteria) we can say:

400 = B₀(1 + r)⁵

Plugging in 50/(1 + r)² for B₀ (from the first equation) we get:

400 = (50/(1 + r)²)(1 + r)⁵

8 = (1 + r)³ then taking the cube root of both sides:

2 = 1 + r

r = 1

Since we know that B₀ = 50/(1 + r)² we can solve for B₀ by plugging in r = 1:

B₀ = 50/(1 + 1)²

B₀ = 50/4

B₀ = 12.5 (which really doesn't make sense because there could not be 12 and 1/2 bacteria at time zero but we'll just go with the math.

So our modeling equation is

B(t) = 12.5(1 + 1)^t or

B(t) = 12.5(2)^t

After 6 hours:, B(6) = 12.5(2)⁶ = 800 bacteria