The population of a culture of bacteria, P(t), where t is time in days, is growing at a rate that is proportional to the population itself and the growth rate is 0.4. The initial population is 10.
What is the population after 30 days?
How long does it take for the population to double?
If you mean k=0.4 in
dP/dt = kP
then
dp/P = 0.4
lnP = 0.4t + c
P = ce^(0.4t)
P(0) = 10 means
P(t) = 10e^(0.4t)
so find P(30)
and solve for t in 10e^(0.4t) = 20
To find the population after 30 days, we can use the formula for exponential growth:
P(t) = P0 * e^(rt)
Where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm, r is the growth rate, and t is the time.
Given that the growth rate, r, is 0.4 and the initial population, P0, is 10, we can substitute these values into the formula.
P(t) = 10 * e^(0.4 * t)
To find the population after 30 days, we substitute t = 30 into the formula.
P(30) = 10 * e^(0.4 * 30)
Using a calculator, we can find that e^(0.4 * 30) is approximately 293.155.
P(30) = 10 * 293.155
P(30) ≈ 2,931.55
Therefore, the population after 30 days is approximately 2,931.55.
To find the time it takes for the population to double, we need to determine when the population P(t) is twice the initial population P0.
2P0 = P0 * e^(rt)
Dividing both sides of the equation by P0:
2 = e^(rt)
Taking the natural logarithm of both sides:
ln(2) = rt
Dividing both sides by r:
t = ln(2) / r
Substituting the growth rate r = 0.4 into the formula:
t = ln(2) / 0.4
Using a calculator, we can find that ln(2) is approximately 0.693.
t ≈ 0.693 / 0.4
t ≈ 1.7325
Therefore, it takes approximately 1.7325 days for the population to double.
To find the population after 30 days and the time it takes for the population to double, we can use the formula for exponential growth:
P(t) = P0 * e^(rt)
where P(t) is the population at time t, P0 is the initial population, e is Euler's number (~2.71828), r is the growth rate, and t is the time.
Given:
P0 = 10 (initial population)
r = 0.4 (growth rate)
t = 30 (time in days)
1. Population after 30 days:
Substituting the given values into the formula, we get:
P(30) = 10 * e^(0.4 * 30)
P(30) ≈ 10 * e^(12)
P(30) ≈ 17445.04 (rounded to two decimal places)
So, the population after 30 days is approximately 17445.04.
2. Time it takes for the population to double:
To find the doubling time, we need to solve the equation P(t) = 2P0, where P(t) is the population at time t.
Substituting the given values, we get:
2P0 = P0 * e^(0.4 * t)
Dividing both sides by P0:
2 = e^(0.4 * t)
Taking the natural logarithm (ln) on both sides:
ln(2) = 0.4 * t
Solving for t:
t = ln(2) / 0.4
t ≈ 1.72 (rounded to two decimal places)
So, it takes approximately 1.72 days for the population to double.