Question

A bacteria culture is started with 200 bacteria. After 4 hours, the population has grown to 668 bacteria. If the population grows exponentially according to the formula Pₜ = P₀(1 + r)t

Find the growth rate. Round your answer to the nearest tenth of a percent.

Answer 1

The growth rate of the bacterial population is calculated to be approximately 13.4% per hour using the exponential growth formula P_t = P_0(1 + r)^t and the given data.

Step-by-step explanation:

To find the growth rate of the bacterial population that grows exponentially, we can use the formula Pt = P0(1 + r)t, where Pt is the final population, P0 is the initial population, r is the growth rate, and t is the time in consistent units (here, hours).

In this case, we have an initial population P0 of 200 bacteria, a final population Pt of 668 bacteria after 4 hours. To solve for r, we rearrange the formula to find r and get:

r = (Pt/P0)1/t - 1

Substituting the provided numbers in the formula:

r = (668/200)1/4 - 1

Calculating this, we get:

r = 1.671/4 - 1 ≈ 0.1340 or 13.40%

Therefore, the growth rate is approximately 13.4% per hour.