A population of a given bacteria is noted at the following intervals.
Hours. 0 1. 9. 12. 18
Population. 400. 520. 4242. 9319. 44982
a) Calculate an algebraic model
b) How long does it take for to population to double?
c) When will the population reach 1 000 000?
clearly, p(t) is exponential
p = a*2^(kt)
at t=0, p=400, so
p(t) = 400*2^(kt)
p(1) = 520, so
400*2^k = 520
k = ln(520/400)/ln2 = 0.3785
At this point, we should really check that the other data points fit this equation, but assuming they do,
(a) p = 400*2^(0.3785t)
(b) t = 1/0.3785 = 2.642 hours
(c) solve 400*2^(0.3785t) = 100000
a) To calculate an algebraic model, we can use exponential growth equation which is given by:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at time t
P0 is the initial population
r is the growth rate
t is the time
Let's calculate the growth rate (r) using the first two data points:
r = (P1 - P0) / P0
= (520 - 400) / 400
= 0.3
Now we can substitute the values of the growth rate (r) and the initial population (P0) into the equation:
P(t) = 400 * (1 + 0.3)^t
b) To find out when the population doubles, we need to solve the equation P(t) = 2 * P0:
2 * 400 = 400 * (1 + 0.3)^t
Dividing both sides by 400, we get:
2 = (1 + 0.3)^t
Taking the logarithm of both sides, we have:
log(2) = t * log(1.3)
Dividing both sides by log(1.3) gives us:
t = log(2) / log(1.3)
Using a calculator, we can find:
t ≈ 4.77 hours
c) To find out when the population reaches 1,000,000, we need to solve the equation P(t) = 1,000,000:
1,000,000 = 400 * (1 + 0.3)^t
Dividing both sides by 400, we get:
2,500 = (1 + 0.3)^t
Taking the logarithm of both sides, we have:
log(2,500) = t * log(1.3)
Dividing both sides by log(1.3) gives us:
t = log(2,500) / log(1.3)
Using a calculator, we can find:
t ≈ 23.12 hours
To calculate an algebraic model for the given population data, we can use the exponential growth model. In the exponential growth model, the population at a given time (t) is given by the formula:
P(t) = P(0) * e^(kt)
Where:
P(t) is the population at time t.
P(0) is the initial population at time t=0.
e is the base of the natural logarithm (approximately 2.71828).
k is the growth rate constant.
To find the values of P(0) and k, we can use the given data points.
a) To calculate an algebraic model:
Using the first data point (t=0, P=400):
400 = P(0) * e^(0k) → P(0) = 400
Using the second data point (t=1, P=520):
520 = 400 * e^(1k)
Dividing the second equation by the first equation:
520/400 = e^(1k)/e^(0k)
1.3 = e^(k)
Now we can solve for k by taking the natural logarithm (ln) of both sides:
ln(1.3) = ln(e^(k))
k ≈ ln(1.3)
Therefore, the algebraic model for the population is:
P(t) = 400 * e^(ln(1.3) * t)
b) To determine how long it takes for the population to double, we need to find the time (t) when P(t) becomes twice the initial population (P(0)).
Let's set up the equation:
2 * P(0) = P(0) * e^(ln(1.3) * t)
Simplifying the equation:
2 = e^(ln(1.3) * t)
Taking the natural logarithm of both sides:
ln(2) = ln(e^(ln(1.3) * t))
Now, isolate the time (t):
t = ln(2) / ln(1.3)
c) To find when the population will reach 1,000,000, we can use the same equation and solve for t:
1,000,000 = 400 * e^(ln(1.3) * t)
Dividing both sides of the equation by 400:
2,500 = e^(ln(1.3) * t)
Taking the natural logarithm of both sides:
ln(2,500) = ln(e^(ln(1.3) * t))
Simplifying and isolating t:
t = ln(2,500) / ln(1.3)
Using a calculator, you can compute the values for t in both cases (doubling time and reaching 1,000,000 population).