A culture started with 2,000 bacteria. After 6 hours, it grew to 2,200 bacteria. Predict how many bacteria will be present after 12 hours. Round to the nearest whole number.
2,420
To predict how many bacteria will be present after 12 hours, we can assume that the bacteria population is growing exponentially. In an exponential growth scenario, the population typically follows the equation P(t) = P₀ * e^(kt), where P(t) represents the population at time t, P₀ represents the initial population, e is Euler's number (approximately 2.71828), k is the growth rate constant, and t is time.
In this case, we can use the population sizes at 6 hours and 12 hours to solve for the growth rate constant.
Step 1: Calculate the growth rate constant (k)
P(t₁) = P₀ * e^(k*t₁)
Where P(t₁) = 2200 (population at 6 hours)
P₀ = 2000 (initial population)
t₁ = 6 hours
2200 = 2000 * e^(k*6)
Divide both sides by 2000:
1.1 = e^(6k)
Take the natural logarithm (ln) of both sides to isolate k:
ln(1.1) = ln(e^(6k))
ln(1.1) = 6k
Solve for k:
k = ln(1.1) / 6
Step 2: Use the growth rate constant to predict the population at 12 hours
P(t₂) = P₀ * e^(kt₂)
Where P(t₂) is the population at 12 hours, t₂ = 12 hours.
P(t₂) = 2000 * e^((ln(1.1) / 6) * 12)
P(t₂) ≈ 2000 * e^(0.0381 * 12)
P(t₂) ≈ 2000 * e^0.4572
P(t₂) ≈ 2000 * 1.579
P(t₂) ≈ 3158
Therefore, we can predict that approximately 3,158 bacteria will be present after 12 hours.