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Answer:
(a) a = 43.25 , b = 41 and c = 1547
(b) The number of craft-beer breweries in 2016 = 4643
Step-by-step explanation:
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, <em>f(0) = </em><em> which implies that </em><em>c = 1547</em><em>.</em>
<em>f(2) = </em><em> = 4a + 2b + c</em>
4a + 2b + c = 1802 ----------- [Equation 1]
<em>f(4) = </em><em> = 16a + 4b + c </em>
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) =
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.
<em>So, f(8) = </em><em> = 4643.</em>