The number of coffee shops in a certain country can be modeled by the quadratic function

f ( x )=110 x squared + 746 x + 4417,

where f(x) is the number of coffee shops and x is the number of years after 2000. Complete parts a and b

a. Find the number of coffee shops in

2004.

b. If the trend described by the model continues, predict the year after 2000 in which the number of coffee shops will be 26,500.

a. The number of coffee shops in 2004
was

The number of coffee shops in the country can be modeled as by the function: f(x) = 110x² + 746x + 4417, where f(x) is the number of coffee shops and x is the number of years after 2000.

a. Now, in 2004, the x value is (2004 - 2000) = 4.

Therefore, f(4) = 110 × 4² + 746 × 4 + 4417 = 9161

So, the number of coffee shops in 2004 is 9161.

b. If f(x) = 26500, then we have to find the value of x.

So, 110x² + 746x + 4417 = 26500

⇒ 110x² + 746x - 22083 = 0

Applying the Sridhar Achaya formula to find x,

[tex]x = \frac{-746 + \sqrt{746^{2} - 4 \times 110 \times (-22083) } }{2 \times 110}[/tex] {Neglecting the negative roots as x can not be negative}

⇒ x = 11.17 years.

If ax² + bx + c = 0, then Using the Sridhar Acharya Formula, we can write

[tex]x = \frac{-b + \sqrt{b^{2} - 4ac } }{2a}[/tex] and [tex]x = \frac{-b - \sqrt{b^{2} - 4ac } }{2a}[/tex]

Therefore, in 2011 the number of coffee shop will be 26500.