Question

12. The table below shows the population of Canada from 2010 to 2019.YearPopulation (millions)201034.0201133.5201234.7201335.1201435.4201535.7201635.1201736.5201837.1201937.6(a) Use a graphing calculator or spreadsheet program to build an exponential regression model to fit this data, letting t=0 in 2010, where Pt is measured in millions of people. Round each coefficient to two decimal places.Pt = (b) What does this model predict that the population of Canada will be in 2035? Round your answer to the nearest tenth (a hundred thousand people). million people(c) When does this model predict that Canada's population will reach 40 million? Give your answer as a calendar year (ex: 2010).During the year

Answer 1

Step 1:

a) From the graphing calculator

$\begin{gathered} \text{The exponential regression model is} \\ y=ab^x \\ a\text{ = }33.67 \\ b\text{ = }1.01 \\ \\ \text{Therefore, y = 33.67 }\times(1.01)^x \\ Pt\text{ = y and x = t} \\ P(t)\text{ = 33.67 }\times1.01^t \end{gathered}$

b)

The population of Canada at 2035, t = 25

$\begin{gathered} P(2035)=33.67\times1.01^{25}^{} \\ =\text{ 33.67 }\times\text{ 1.2824} \\ =\text{ 43.2 million} \end{gathered}$

c)

P(t) = 40

t = ?

$\begin{gathered} P(t)\text{ = 33.67 }\times1.01^t \\ 40\text{ = 33.67 }\times1.01^t \\ 1.01^t\text{ = }\frac{40}{33.67} \\ 1.01^t\text{ = 1.188} \\ \text{Take log of both sides} \\ \log ^{1.01^t}_{}\text{ = }\log ^{1.188}_{} \\ t\text{ }\log ^{1.01}_{}\text{ = }\log ^{1.188}_{} \\ t\text{ = }\frac{\log ^{1.188}_{}}{\log ^{1.01}_{}} \\ t\text{ = }\frac{0.074816}{0.00432} \\ t\text{ = 17.32} \end{gathered}$

t = 17 years

The model predicts Canada will reach 40 million in 2027