SOLUTION: A bacteria culture grows with constant relative growth rate .the bacteria count was 200 after 2 hours and 14000 was 5 hours .what is the relative growth rate .what is the initial s

Question 1185949: A bacteria culture grows with constant relative growth rate .the bacteria count was 200 after 2 hours and 14000 was 5 hours .what is the relative growth rate .what is the initial size of culture . Find the number of cells after 4.5 hours. Find the rate growth after 5.5 hours and when will the population reach 25 000
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to solve this exponential growth problem:
**1. Set up the exponential growth formula:**
The formula for population growth is: P(t) = P₀ * e^(rt)
Where:
* P(t) is the population at time t
* P₀ is the initial population
* r is the relative growth rate
* t is the time
**2. Use the given information to create two equations:**
* Equation 1: 200 = P₀ * e^(2r) (after 2 hours)
* Equation 2: 14000 = P₀ * e^(5r) (after 5 hours)
**3. Solve for r (relative growth rate):**
Divide Equation 2 by Equation 1:
14000 / 200 = (P₀ * e^(5r)) / (P₀ * e^(2r))
70 = e^(3r)
Take the natural logarithm of both sides:
ln(70) = 3r
r = ln(70) / 3
r ≈ 1.424 / 3
r ≈ 0.475 (or 47.5%)
**4. Solve for P₀ (initial size of the culture):**
Substitute the value of r back into either Equation 1 or Equation 2. Let's use Equation 1:
200 = P₀ * e^(2 * 0.475)
200 = P₀ * e^0.95
P₀ = 200 / e^0.95
P₀ ≈ 200 / 2.586
P₀ ≈ 77
**5. Find the number of cells after 4.5 hours:**
P(4.5) = 77 * e^(0.475 * 4.5)
P(4.5) ≈ 77 * e^2.1375
P(4.5) ≈ 77 * 8.468
P(4.5) ≈ 652
**6. Find the rate of growth after 5.5 hours:**
The rate of growth is given by the derivative of the population function:
dP/dt = r * P(t)
First, find the population at t = 5.5:
P(5.5) = 77 * e^(0.475 * 5.5) ≈ 77 * 11.85 ≈ 912.45
Now, find the rate of growth:
dP/dt = 0.475 * 912.45 ≈ 434 bacteria/hour
**7. When will the population reach 25,000?**
25000 = 77 * e^(0.475t)
25000/77 = e^(0.475t)
ln(25000/77) = 0.475t
t = ln(25000/77) / 0.475
t ≈ ln(324.675) / 0.475
t ≈ 5.78 / 0.475
t ≈ 12.17 hours
**Summary of Answers:**
* Relative growth rate (r) ≈ 47.5% or 0.475
* Initial size of the culture (P₀) ≈ 77
* Number of cells after 4.5 hours ≈ 652
* Rate of growth after 5.5 hours ≈ 434 bacteria/hour
* Time to reach 25,000 cells ≈ 12.17 hours

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