The Problem:

The population of Manitoba from 1975 to 2010 is given in 5-year increments.

Year/Population
1975
1,019,529
1980
1,033,557
1985
1,078,327
1990
1,102,752
1995
1,124,947
2000
1,144,428
2005
1,176,114
2010
1,226,738


1. Find a quadratic equation of best fit.
2. Find a cubic equation of best fit.
3. Find an exponential equation of best fit.
4. Using mathematical reasoning,state which model you think best suits the data.
5. Use that model to predict the population in 2050.
6. State at least two factors that affect population growth.

1 mark for a correct quadratic model
1 mark for a correct cubic model
1 mark for a correct exponential model
1 mark for a correct justification of choice of model
1 mark for correct population in 2050
1 mark for showing work to find population in 2050
2 marks for population factors

Please go back and post what you know about the other questions you've posted.

Show a little effort on your part and tutors will be more willing to help you.

Do not post any new questions until you've done that.

Lol just answer the question or go away thanks

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To answer the questions and solve the problem, we will use mathematical modeling techniques to find the best-fitting equations and make predictions. I will explain how to approach each question step-by-step.

1. Find a quadratic equation of best fit:
To find a quadratic equation that best fits the given data points, we can use the method of least squares. The quadratic equation can be written as:
y = ax^2 + bx + c

Using the given data points, we can set up a system of three equations and solve for the coefficients a, b, and c. The three equations are obtained by substituting the year (x) and the corresponding population (y) values from three data points into the equation.

For example, using the data points (1975, 1,019,529), (1980, 1,033,557), and (1985, 1,078,327), we get the following equations:

1975: 1,019,529 = a(1975)^2 + b(1975) + c
1980: 1,033,557 = a(1980)^2 + b(1980) + c
1985: 1,078,327 = a(1985)^2 + b(1985) + c

Solving this system of equations will give us the values of a, b, and c, and we can substitute them back into the quadratic equation to get the equation of best fit.

2. Find a cubic equation of best fit:
Similarly, we can find a cubic equation that best fits the given data using the method of least squares. The cubic equation can be written as:
y = ax^3 + bx^2 + cx + d

Following the same approach as in the quadratic case, we can set up a system of four equations using four data points and solve for the coefficients a, b, c, and d.

3. Find an exponential equation of best fit:
To find an exponential equation of best fit, we need to take the logarithm of both sides of the equation. The exponential equation can be written in the form:
y = ab^x

By taking the logarithm on both sides, we can convert this exponential equation into a linear equation of the form:
log(y) = log(a) + xlog(b)

Now, we can treat log(y) as the dependent variable and x as the independent variable and use linear regression techniques to find the equation of best fit.

4. Using mathematical reasoning, state which model you think best suits the data:
After obtaining the equations of best fit for quadratic, cubic, and exponential models, we need to analyze the goodness of fit for each model. This can be done by calculating the coefficient of determination (R^2) for each model. The R^2 value measures how well the model explains the variation in the data. A higher R^2 value indicates a better fit.

Compare the R^2 values for the quadratic, cubic, and exponential models. Based on the R^2 value, we can justify which model fits the data best. The model with the highest R^2 value can be considered the best fit.

5. Use that model to predict the population in 2050:
Once we have the best-fit model, we can use it to predict the population in 2050. Substitute x = 2050 into the equation, and calculate the corresponding value of y to get the predicted population in that year.

6. State at least two factors that affect population growth:
Factors that affect population growth can include birth rates, death rates, immigration, emigration, availability of resources, economic conditions, government policies, and social factors. You can state any two factors that you believe have a significant influence on population growth based on your knowledge or research.

By following these steps and performing the necessary calculations, you will be able to answer the given set of questions and solve the problem.