Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields 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 choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.