The population P of a particular city, Metropia, is growing at a rate proportional to the
current population.
The population at time t years is modelled by the equation P = Aekt where A and k are
constants.
(a) With the aid of appropriate examples, explain how the growth of P over time
would be influenced by:
(i) The magnitude of A
(ii) The magnitude of k
(iii) The sign of k
(b) If the population of Metropia was 1,000,000 in 2005 (at t = 0) and 1,072,500
in 2007, determine the values of A and k and hence write an equation to model
Metropia’s population with respect to time.
(c) Predict the population of Metropia in the year 2012.
(d) Determine the expected length of time it would take for the population of Metropia
to double.
What is your question about this?
(a)
(i) The magnitude of A:
The constant A represents the initial population of Metropia at time t = 0. The larger the value of A, the larger the initial population. So, the magnitude of A directly influences the starting point of population growth. For example, if A is 100,000, the initial population is 100,000. If A is 1,000,000, the initial population is 1,000,000.
(ii) The magnitude of k:
The constant k represents the growth rate of the population. A larger value of k indicates a faster population growth rate, while a smaller value of k indicates a slower growth rate. For example, if k is 0.05, the population will grow slowly over time. If k is 0.2, the population will grow at a faster rate.
(iii) The sign of k:
The sign of k indicates whether the population is growing or declining. If k is positive, the population is growing exponentially. If k is negative, the population is declining exponentially. For example, if k is positive, the population will increase over time. If k is negative, the population will decrease over time.
(b)
To determine the values of A and k, we can use the given population data for 2005 and 2007:
At t = 0 (year 2005): P = 1,000,000
At t = 2 (year 2007): P = 1,072,500
Substituting these values into the equation P = Aekt, we get:
1,000,000 = Ae^(0) --> A = 1,000,000
1,072,500 = 1,000,000e^(2k) --> e^(2k) = 1.0725
To solve for k, take the natural logarithm of both sides:
ln(e^(2k)) = ln(1.0725)
2k = ln(1.0725)
k = ln(1.0725) / 2
Therefore, A = 1,000,000 and k ≈ 0.035.
The equation to model Metropia's population with respect to time is: P = 1,000,000e^(0.035t).
(c)
To predict the population in the year 2012, we need to find the value of P at t = 2012 - 2005 = 7 years.
P = 1,000,000e^(0.035 * 7)
Calculate the value using a calculator or software.
(d)
To determine the expected length of time it would take for the population of Metropia to double, we need to find when P = 2 * 1,000,000:
2,000,000 = 1,000,000e^(0.035t)
Divide both sides by 1,000,000 and take the natural logarithm:
2 = e^(0.035t)
To solve for t, take the natural logarithm of both sides:
ln(2) = ln(e^(0.035t))
ln(2) = 0.035t * ln(e)
t = ln(2) / 0.035
Therefore, the expected length of time for the population of Metropia to double is approximately t ≈ ln(2) / 0.035 years.