Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Math problem?

You are the housing minister in the year 1996 for a country with 30 million people. you have been asked to predict the population 5 years from 1996. as part of a 10 year master plan for housing. census records show that the population was 22.684 million in 1986 and 26.087 in 1991. Give your prediction, with justification in words

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There are a few ways you can look at this. Since we have three points on this curve, we can set up quadratic with these three points to end up with a system of three equations and three unknowns that we can solve for.  There are other possible models that we can set up, but this is the one I'll choose (it probably also depends on what type of math class you are in and what you know, etc.)

If we set up this equation:

y = ax² + bx + c

and call x the number of years after 1996 and y the populations in millions, we have the following three points:

(-10, 22.684)

(-5, 26.087)

(0, 30)

Putting these values into the general equation gives us these three equations:

22.684 = a(-10)² + b(-10) + c and 26.087 = a(-5)² + b(-5) + c and 30 = a(0)² + b(0) + c

Simplifying them, we get a solution for c which then we can substitute into the other two equations:

22.684 = 100a - 10b + c and 26.087 = 25a - 5b + c and 30 = 0 + 0 + c

22.684 = 100a - 10b + c and 26.087 = 25a - 5b + c and 30 = c

22.684 = 100a - 10b + 30 and 26.087 = 25a - 5b + 30

-7.316 = 100a - 10b and -3.913 = 25a - 5b

We can solve both equations for b in terms of a, then substituting them into each other to get one equation with one unknown that we can solve:

-7.316 - 100a = -10b and -3.913 - 25a = -5b

0.7316 + 10a = b and 0.7826 + 5a = b

0.7316 + 10a = 0.7826 + 5a

5a = 0.051

a = 0.0102

Now that we have "a", we can solve for "b":

b = 0.7826 + 5a

b = 0.7826 + 5(0.0102)

b = 0.7826 + 0.051

b = 0.8336

so our equation is now:

f(x) = 0.0102x² + 0.8336x + 30

What would the population be in 5 years?  Solve for f(5):

f(5) = 0.0102(5)² + 0.8336(5) + 30

f(5) = 0.0102(25) + 0.8336(5) + 30

f(5) = 0.255 + 4.168 + 30

f(5) = 34.423

The population could be 34.423 million in the year 2001, if this model is the correct one.   Again, multiple models could exist which would give you different answers.

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• it is definitely 8

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