To solve this problem, we need to derive the revenue function [tex]\( R(x) \)[/tex] in terms of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] is the number of [tex]\( \$3 \)[/tex] decreases in the ticket price.
1. Determine new ticket price and new number of tickets sold:
- Initial ticket price: \[tex]$75
- Decrease per price drop: \$[/tex]3
- Initial average ticket sales: 340 tickets
- Increase in sales per price drop: 4 tickets
After [tex]\( x \)[/tex] price decreases:
[tex]\[
\text{New price} = 75 - 3x \quad (\text{each } x \text{ decreases the price by } \$3)
\][/tex]
[tex]\[
\text{New number of tickets sold} = 340 + 4x \quad (\text{each } x \text{ increases the ticket sales by 4 tickets})
\][/tex]
2. Formulate the revenue:
Revenue [tex]\( R(x) \)[/tex] is the product of the new ticket price and the new number of tickets sold.
[tex]\[
R(x) = (\text{New price}) \times (\text{New number of tickets sold})
\][/tex]
Substituting the expressions derived above:
[tex]\[
R(x) = (75 - 3x)(340 + 4x)
\][/tex]
3. Expand the revenue formula:
[tex]\[
R(x) = (75 \cdot 340) + (75 \cdot 4x) - (3x \cdot 340) - (3x \cdot 4x)
\][/tex]
Let's expand each term:
[tex]\[
75 \cdot 340 = 25,500
\][/tex]
[tex]\[
75 \cdot 4x = 300x
\][/tex]
[tex]\[
3x \cdot 340 = 1020x
\][/tex]
[tex]\[
3x \cdot 4x = 12x^2
\][/tex]
Substituting these values back into the equation:
[tex]\[
R(x) = 25,500 + 300x - 1020x - 12x^2
\][/tex]
Combine like terms:
[tex]\[
R(x) = 25,500 - 720x - 12x^2
\][/tex]
4. Identify the correct option:
The derived revenue function is:
[tex]\[
R(x) = 25,500 - 720x - 12x^2
\][/tex]
Hence, the correct answer is:
C. [tex]\( R(x) = 25,500 - 720x - 12x^2 \)[/tex]
