U.S. craft-beer breweries (breweries that make fewer than 6 million barrels annually and are less than 25% owned by big breweries) have been doing a booming business. The number of these small breweries from 2008 through 2012 can be modeled by using a quadratic function of the form f(t) = at² + bt + c where a, b, and c are constants and t is measured in years, with t = 0 corresponding to 2008.
(a) Find a, b, and c if f(0) = 1547, f(2) = 1802, and f(4) = 2403.
(b) Use the model obtained in part (a) to estimate the number of craft-beer breweries in 2016, assuming that the trend continued. (Round your answer to the nearest integer.)

Given that the number of small craft-beer breweries from 2008 to 2012 is modeled by using a quadratic function of the form f(t) = at² + bt + c where a, b, and c are constants and t is measured in years, with t = 0 corresponding to 2008.

(a) Now we have to find these constants given f(0) = 1547, f(2) = 1802, and f(4) = 2403.

So, f(0) = [tex]a*(0)^{2} + b*(0) + c[/tex] which implies that c = 1547.

f(2) = [tex]a*(2)^{2} + b*(2) + c[/tex] = 4a + 2b + c

4a + 2b + c = 1802 ----------- [Equation 1]

f(4) = [tex]a*(4)^{2} + b*(4) + c[/tex] = 16a + 4b + c

16a + 4b + c = 2403 ----------- [Equation 2]

Further solving Equation 1 we get; 4a + 2b + 1547 = 1802

4a + 2b = 1802 - 1547 ; 4a + 2b = 255 ----------- [Equation 3]

Further solving Equation 2 we get; 16a + 4b + 1547 = 2403

16a + 4b = 2403 - 1547 ; 16a + 4b = 856 ----------- [Equation 4]

Now, Solving Equation 3 and 4 we get ;

a = 43.25 and b = 41 .

Therefore, values of constants a, b and c are 43.25, 41 and 1547 respectively.

(b) So, model we obtained in part (a) is represented as :

f(t) = [tex]43.25t^{2} + 41t + 1547[/tex]

Now to estimate the number of craft-beer breweries in 2016, assuming that the trend continued we know that t would be 8 as 2016 - 2008 = 8 years.

So, f(8) = [tex]43.25*(8)^{2} + 41*8 + 1547[/tex] = 4643.