a bacteria culture grows with a constant per capita growth rate. After 2 hours there are 500 bacteria and after 6 hours the count is 312500, find the initial population and the population after 8 hours
To find the initial population, we can use the exponential growth formula:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at a given time t
P0 is the initial population
r is the per capita growth rate
t is the time
Let's use the information given to solve for P0:
At t = 2 hours, P(2) = 500
At t = 6 hours, P(6) = 312500
Using the formula for each of these time points, we get two equations:
Equation 1: 500 = P0 * (1 + r)^2
Equation 2: 312500 = P0 * (1 + r)^6
We now have a system of equations with two unknowns (P0 and r). To solve it, we can divide Equation 2 by Equation 1:
312500/500 = (P0*(1+r)^6) / (P0*(1+r)^2)
625 = (1 + r)^4
Take the fourth root of both sides:
(1 + r) = ∛625 = 5
Solve for r:
1 + r = 5
r = 5 - 1
r = 4
Now we can substitute the value of r back into Equation 1 to solve for P0:
500 = P0 * (1 + 4)^2
500 = P0 * (5)^2
500 = 25P0
P0 = 20
Therefore, the initial population is 20.
To find the population after 8 hours, we can substitute P0 and r into the exponential growth formula:
P(8) = 20 * (1 + 4)^8
P(8) = 20 * (5)^8
P(8) = 20 * 390625
P(8) = 7812500
Therefore, the population after 8 hours is 7,812,500.
To find the initial population, we can set up an exponential growth equation.
Let P(t) represent the population at time t, and r be the per capita growth rate.
The exponential growth equation is given by:
P(t) = P₀ * e^(r*t)
Given that after 2 hours there are 500 bacteria, we can write:
500 = P₀ * e^(r*2) --(1)
Similarly, after 6 hours, the count is 312500:
312500 = P₀ * e^(r*6) --(2)
To find the initial population (P₀), we need to solve equations (1) and (2) simultaneously.
Dividing equation (2) by equation (1) gives:
312500/500 = (P₀ * e^(r*6)) / (P₀ * e^(r*2))
625 = e^(r*6 - r*2)
625 = e^(4r)
To solve for r, we take the natural logarithm (ln) of both sides:
ln 625 = ln e^(4r)
ln 625 = 4r * ln e
ln 625 = 4r
Now, we can solve for r by dividing both sides by 4:
r = ln 625 / 4
Using a calculator, we find that r ≈ 0.5878.
Substituting the value of r back into equation (1), we can solve for P₀:
500 = P₀ * e^(0.5878*2)
500 = P₀ * e^(1.1756)
500 = P₀ * 3.2417
P₀ = 500 / 3.2417
P₀ ≈ 154.25
Therefore, the initial population is approximately 154 bacteria.
To find the population after 8 hours, we can use the equation P(t) = P₀ * e^(r*t):
P(8) = 154.25 * e^(0.5878*8)
P(8) = 154.25 * e^(4.7024)
P(8) ≈ 154.25 * 110.1517
P(8) ≈ 16943
Therefore, the population after 8 hours is approximately 16943 bacteria.