The number of franchises of a popular cafe has been growing since the store opened in 1971. Since then, the number of stores grew at a rate of 33% per year.

a) Explain how you could create an algebraic model that gives the number of stores in any year after 1971. How the information in the problem relates to your algebraic model.

b) Use your model to predict the number of stores in 2010.

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The number of franchises of the popular cafe has been growing at a rate of 33% every year since the first store was opened in 1971. Taking the growth to be compounded every year gives the number of stores in any year t after the year 1971 as `N(t) = 1*(1.33)^t`

The general equation of compounded growth is `x(t) = xo*(1+r)^t` , from the information in the problem, `xo = 1 and r = 0.33`

The year 2010 is 39 years after 1971. From the equation derived earlier, the number of stores is `1*(1.33)^39 = 67641`

**The required equation is `N(t) = 1*(1.33)^t` and the number of stores in the year 2010 is 67641**

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