Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.