To find the correct equation, let's first set up the relationship between the ticket price, the number of tickets sold, and the revenue.
1. Let [tex]\( x \)[/tex] be the number of [tex]$\$[/tex] 0.25[tex]$ increases.
2. The new ticket price after \( x \) increases is:
\[ 8.50 + 0.25x \]
3. The new number of tickets sold after \( x \) increases is:
\[ 200 - 5x \]
4. The revenue \( R \) can be represented as:
\[ R = \text{(new ticket price)} \times \text{(new number of tickets sold)} \]
\[ R = (8.50 + 0.25x) \times (200 - 5x) \]
5. We are told the target revenue is \$[/tex]1,700, so we can set up the equation:
[tex]\[ (8.50 + 0.25x)(200 - 5x) = 1700 \][/tex]
Let's expand and simplify this equation step by step:
6. Distribute the terms inside the parentheses:
[tex]\[ 8.50 \cdot 200 + 8.50 \cdot (-5x) + 0.25x \cdot 200 + 0.25x \cdot (-5x) = 1700 \][/tex]
[tex]\[ 1700 - 42.5x + 50x - 1.25x^2 = 1700 \][/tex]
7. Combine the like terms:
[tex]\[ 1700 + (50x - 42.5x) - 1.25x^2 = 1700 \][/tex]
[tex]\[ 1700 + 7.5x - 1.25x^2 = 1700 \][/tex]
8. Subtract 1700 from both sides to simplify:
[tex]\[ 0 + 7.5x - 1.25x^2 = 0 \][/tex]
[tex]\[ -1.25x^2 + 7.5x = 0 \][/tex]
So, the equation that the company needs to solve to find the number of price increases [tex]\( x \)[/tex] is:
[tex]\[ -1.25x^2 + 7.5x = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
