Answer :
To determine the correct equation that the theater company could solve to find the number of price increases [tex]\( x \)[/tex] and still achieve a revenue of \[tex]$1,700, let's break down the problem step-by-step:
1. Identify the initial conditions:
- Initial ticket price: \$[/tex]8.50
- Initial tickets sold: 200
2. Determine the changes per increment:
- Price increase per increment: \[tex]$0.25 - Decrease in ticket sales per increment: 5 tickets 3. Set up the revenue equation: The revenue \( R \) is the product of the price per ticket and the number of tickets sold. If \( x \) is the number of price increments of \$[/tex]0.25, the new ticket price becomes [tex]\( \$8.50 + \$0.25 \times x \)[/tex] and the new number of tickets sold is [tex]\( 200 - 5 \times x \)[/tex].
So, the revenue equation is:
[tex]\[ R = (\text{Initial price} + \text{price increase per increment} \times x) \times (\text{Initial tickets sold} - \text{decrease in ticket sales per increment} \times x) \][/tex]
Given the revenue is \$1,700, we substitute [tex]\( R = 1700 \)[/tex]:
[tex]\[ 1700 = (8.50 + 0.25x)(200 - 5x) \][/tex]
4. Expand and simplify the equation:
First, expand the right side:
[tex]\[ 1700 = (8.50 \times 200) + (8.50 \times -5x) + (0.25x \times 200) + (0.25x \times -5x) \][/tex]
Simplify each term:
[tex]\[ 1700 = 1700 - 42.5x + 50x - 1.25x^2 \][/tex]
Combine like terms:
[tex]\[ 1700 = 1700 + 7.5x - 1.25x^2 \][/tex]
5. Move all terms to one side to form a quadratic equation:
[tex]\[ 1700 - 1700 = -1.25x^2 + 7.5x \][/tex]
Simplify the equation:
[tex]\[ 0 = -1.25x^2 + 7.5x \][/tex]
6. Compare with the given choices:
- Choice A: [tex]\(-1.25 x^2 - 7.5 x - 1,700 = 0\)[/tex]
- Choice B: [tex]\(-1.25 x^2 - 7.5 x = 0\)[/tex]
- Choice C: [tex]\(-1.25 x^2 + 7.5 x = 0\)[/tex]
- Choice D: [tex]\(-1.25 x^2 + 7.5 x - 1,700 = 0\)[/tex]
The correct equation is:
[tex]\[ 0 = -1.25x^2 + 7.5x \][/tex]
This matches choice B.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
- Initial tickets sold: 200
2. Determine the changes per increment:
- Price increase per increment: \[tex]$0.25 - Decrease in ticket sales per increment: 5 tickets 3. Set up the revenue equation: The revenue \( R \) is the product of the price per ticket and the number of tickets sold. If \( x \) is the number of price increments of \$[/tex]0.25, the new ticket price becomes [tex]\( \$8.50 + \$0.25 \times x \)[/tex] and the new number of tickets sold is [tex]\( 200 - 5 \times x \)[/tex].
So, the revenue equation is:
[tex]\[ R = (\text{Initial price} + \text{price increase per increment} \times x) \times (\text{Initial tickets sold} - \text{decrease in ticket sales per increment} \times x) \][/tex]
Given the revenue is \$1,700, we substitute [tex]\( R = 1700 \)[/tex]:
[tex]\[ 1700 = (8.50 + 0.25x)(200 - 5x) \][/tex]
4. Expand and simplify the equation:
First, expand the right side:
[tex]\[ 1700 = (8.50 \times 200) + (8.50 \times -5x) + (0.25x \times 200) + (0.25x \times -5x) \][/tex]
Simplify each term:
[tex]\[ 1700 = 1700 - 42.5x + 50x - 1.25x^2 \][/tex]
Combine like terms:
[tex]\[ 1700 = 1700 + 7.5x - 1.25x^2 \][/tex]
5. Move all terms to one side to form a quadratic equation:
[tex]\[ 1700 - 1700 = -1.25x^2 + 7.5x \][/tex]
Simplify the equation:
[tex]\[ 0 = -1.25x^2 + 7.5x \][/tex]
6. Compare with the given choices:
- Choice A: [tex]\(-1.25 x^2 - 7.5 x - 1,700 = 0\)[/tex]
- Choice B: [tex]\(-1.25 x^2 - 7.5 x = 0\)[/tex]
- Choice C: [tex]\(-1.25 x^2 + 7.5 x = 0\)[/tex]
- Choice D: [tex]\(-1.25 x^2 + 7.5 x - 1,700 = 0\)[/tex]
The correct equation is:
[tex]\[ 0 = -1.25x^2 + 7.5x \][/tex]
This matches choice B.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
