The cost​ (in millions of​ dollars) for a​ 30-second ad during the TV broadcast of a major sporting event can be approximated by the rational expression StartFraction 0.499 x squared minus 4.326 x plus 29.7 Over x plus 4 EndFraction
​, where xequals6 corresponds to 2006.
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x equals 14

An ad in 2014 cost approximately ​$__.?
​(Do not round until the final answer. Then round to the nearest hundredth as​ needed.)

Given:

\[
C(x) = \frac{0.499x^2 - 4.326x + 29.7}{x + 4}
\]

Where $x = 6$ corresponds to the year 2006. For 2014, $x = 14$. Let's calculate the cost using this expression.

1. Substitute $x = 14$ into the expression:

\[
C(14) = \frac{0.499(14)^2 - 4.326(14) + 29.7}{14 + 4}
\]

2. Compute each part:
   - $14^2 = 196$
   - $0.499 \times 196 = 97.804$
   - $4.326 \times 14 = 60.564$
   - Numerator: $97.804 - 60.564 + 29.7$
   - Denominator: $14 + 4 = 18$

3. Combine and simplify:

\[
C(14) = \frac{97.804 - 60.564 + 29.7}{18}
\]

\[
C(14) = \frac{66.94}{18} \approx 3.719
\]

Therefore, the cost of the ad in 2014 was approximately **$3.72 million**.

Would you like further details or have any questions? Here are some related questions to explore:

1. How would the cost change in 2020 (i.e., when $x = 20$) using this model?
2. What year corresponds to $x = 0$ in this model?
3. How does the cost change annually according to this rational expression?
4. What are the implications of the model's numerator and denominator in terms of cost predictions?
5. What is the rate of change of the cost at $x = 14$?

**Tip:** Always substitute the values into the expression accurately before simplifying for precise results.

The cost (in millions of dollars) for a 30-second ad during the TV broadcast of a major sporting event can be approximated by the rational expression (0.499x^2 - 4.326x + 29.7) / (x + 4), where x=6 corresponds to 2006. An ad in 2014 (x=14) cost approximately $__.?

This problem involves calculating the cost of a 30-second ad in 2014 using the rational expression (0.499x^2 - 4.326x + 29.7) / (x + 4). The problem covers key algebraic concepts, including rational expressions and quadratic functions. Suitable for students in Grades 9-12.

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