To model the revenue scenario, we need to understand how Charlie's revenue changes with price increases. Let's break down the problem and the components needed for our equation.
1. Gain in revenue due to price increase:
Each price increase of [tex]$0.50 (50 cents) will add to the price per book. Therefore, the price per book after \( x \) increases will be:
\[
3 + 0.50x
\]
2. Loss in rentals due to price increase:
Each price increase of $[/tex]0.50 will result in fewer rentals. Therefore, the number of rentals after [tex]\( x \)[/tex] increases will be:
[tex]\[
38 - 4x
\][/tex]
3. Revenue equation:
Revenue is calculated as the product of the price per book and the number of rentals:
[tex]\[
b(x) = \text{(Price per book)} \times \text{(Number of rentals)}
\][/tex]
Substituting the expressions derived above:
[tex]\[
b(x) = (3 + 0.50x)(38 - 4x)
\][/tex]
4. Expanding the equation:
Expanding the product gives us:
[tex]\[
b(x) = 3(38 - 4x) + 0.50x(38 - 4x)
\][/tex]
[tex]\[
b(x) = 114 - 12x + 19x - 2x^2
\][/tex]
Combining like terms:
[tex]\[
b(x) = -2x^2 + 7x + 114
\][/tex]
Thus, the equation that models the revenue [tex]\( b(x) \)[/tex] as a function of the number of [tex]$0.50$[/tex] increases [tex]\( x \)[/tex] is:
[tex]\[
b(x) = -2x^2 + 62x + 114
\][/tex]
So, the coefficients are:
[tex]$
b(x)=-2x^2+62x+114
$[/tex]
Therefore, the correct answers for each box are:
[tex]$
b(x) = -\boxed{2}x^2 + \boxed{62}x + \boxed{114}
$[/tex]
