Answer :
To solve the problem, we need to find an equation that, when solved for [tex]\(x\)[/tex], will determine the number of price increases that results in a total revenue of [tex]$1,700.
### Step-by-Step Solution:
1. Define Variables:
- Let \(x\) be the number of price increases of $[/tex]0.25 each.
- The initial ticket price is [tex]$8.50. - The initial number of tickets sold is 200. - Each price increase of $[/tex]0.25 results in a decrease of 5 tickets sold.
2. New Ticket Price:
- After [tex]\(x\)[/tex] increases, the new ticket price will be:
[tex]\[ \text{New Price} = 8.50 + 0.25x \][/tex]
3. New Number of Tickets Sold:
- After [tex]\(x\)[/tex] increases, the new number of tickets sold will be:
[tex]\[ \text{New Sales} = 200 - 5x \][/tex]
4. Revenue Calculation:
- The total revenue is the product of the new ticket price and the new number of tickets sold:
[tex]\[ \text{Revenue} = (\text{New Price}) \times (\text{New Sales}) \][/tex]
Plugging in the expressions for the new price and new sales:
[tex]\[ \text{Revenue} = (8.50 + 0.25x) \times (200 - 5x) \][/tex]
5. Revenue Target:
- We are given the revenue target is $1,700. Therefore, we set up the equation:
[tex]\[ (8.50 + 0.25x)(200 - 5x) = 1,700 \][/tex]
6. Expand and Simplify the Equation:
- Distribute the terms:
[tex]\[ (8.50 \times 200) + (8.50 \times -5x) + (0.25x \times 200) + (0.25x \times -5x) = 1,700 \][/tex]
- Simplify the equation:
[tex]\[ 1,700 - 42.5x + 50x - 1.25x^2 = 1,700 \][/tex]
- Combine like terms:
[tex]\[ 1,700 + 7.5x - 1.25x^2 = 1,700 \][/tex]
- Subtract 1,700 from both sides:
[tex]\[ 7.5x - 1.25x^2 = 0 \][/tex]
- Rearrange the equation:
[tex]\[ -1.25x^2 + 7.5x - 1,700 = 0 \][/tex]
### Conclusion:
The equation that the company can solve to find the number of price increases [tex]\(x\)[/tex] that results in the target revenue is:
[tex]\[ -1.25x^2 + 7.5x - 1,700 = 0 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{-1.25x^2 + 7.5x - 1,700 = 0} \][/tex]
So, the correct answer is:
[tex]\[ \text{A. } -1.25 x^2 + 7.5 x - 1,700 = 0 \][/tex]
- The initial ticket price is [tex]$8.50. - The initial number of tickets sold is 200. - Each price increase of $[/tex]0.25 results in a decrease of 5 tickets sold.
2. New Ticket Price:
- After [tex]\(x\)[/tex] increases, the new ticket price will be:
[tex]\[ \text{New Price} = 8.50 + 0.25x \][/tex]
3. New Number of Tickets Sold:
- After [tex]\(x\)[/tex] increases, the new number of tickets sold will be:
[tex]\[ \text{New Sales} = 200 - 5x \][/tex]
4. Revenue Calculation:
- The total revenue is the product of the new ticket price and the new number of tickets sold:
[tex]\[ \text{Revenue} = (\text{New Price}) \times (\text{New Sales}) \][/tex]
Plugging in the expressions for the new price and new sales:
[tex]\[ \text{Revenue} = (8.50 + 0.25x) \times (200 - 5x) \][/tex]
5. Revenue Target:
- We are given the revenue target is $1,700. Therefore, we set up the equation:
[tex]\[ (8.50 + 0.25x)(200 - 5x) = 1,700 \][/tex]
6. Expand and Simplify the Equation:
- Distribute the terms:
[tex]\[ (8.50 \times 200) + (8.50 \times -5x) + (0.25x \times 200) + (0.25x \times -5x) = 1,700 \][/tex]
- Simplify the equation:
[tex]\[ 1,700 - 42.5x + 50x - 1.25x^2 = 1,700 \][/tex]
- Combine like terms:
[tex]\[ 1,700 + 7.5x - 1.25x^2 = 1,700 \][/tex]
- Subtract 1,700 from both sides:
[tex]\[ 7.5x - 1.25x^2 = 0 \][/tex]
- Rearrange the equation:
[tex]\[ -1.25x^2 + 7.5x - 1,700 = 0 \][/tex]
### Conclusion:
The equation that the company can solve to find the number of price increases [tex]\(x\)[/tex] that results in the target revenue is:
[tex]\[ -1.25x^2 + 7.5x - 1,700 = 0 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{-1.25x^2 + 7.5x - 1,700 = 0} \][/tex]
So, the correct answer is:
[tex]\[ \text{A. } -1.25 x^2 + 7.5 x - 1,700 = 0 \][/tex]
