Question

Modeling London's

Population
Task
The table below shows historical estimates for the
population of London.
Year
1801
1821 1841 1861 1881
1901 1921 1939
1961
London
population 1,100,000 1,600,000 2,200,000 3,200,000 4,700,000 6,500,000 7,400,000 8,600,000 8,000,000
No data was available in 1941 because of the war.
a. Can the London population data be accurately
modeled by a linear, quadratic, or exponential function?
Explain.
b. A logistic growth equation can be written in the form
a
P(1) = 1 + e-b(t-c)
where a, b, and care positive numbers and t represents
time measured in years. Using the application supplied,
determine if the London population data can be
accurately modeled by a logistic equation.
a
c. Explain the shape of the graph of P in terms of the
structure of the equation P(t) = 1). What impact do
the values of a, b, and c have on the graph of P?

Answer 1

a. The London population data cannot be accurately modeled by a linear function because the population growth is not constant over time. A quadratic function may be able to fit the data, but it is unlikely to accurately model the population growth in the long term. An exponential function is a more suitable choice because it can model exponential population growth, which is common in real-world scenarios.

b. Using a logistic growth equation, we can determine if the London population data can be accurately modeled by a logistic equation. We can use a regression analysis to find the values of a, b, and c that best fit the data, and then compare the predicted values to the actual population data to determine the accuracy of the model.

c. The logistic growth equation P(t) = 1 / (a + be^(-b(t-c))) produces an S-shaped curve, which represents the growth of a population that is initially slow, then accelerates, and finally levels off as it approaches a carrying capacity. The value of a represents the initial population size, b represents the growth rate, and c represents the time at which the population growth rate is highest. As a increases, the curve shifts upwards, indicating a larger initial population size. As b increases, the curve becomes steeper, indicating faster population growth. As c increases, the curve shifts to the right, indicating a later time at which the population growth rate is highest.