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A theater company is considering raising the price of its tickets. It currently charges [tex]$\$ 8.50$[/tex] for each ticket and sells an average of 200 tickets for each show. The company estimates that for each [tex]$\[tex]$ 0.25$[/tex][/tex] increase in the price of the ticket, the average ticket sales will go down by 5 people.

Which equation could the company solve to find the number of price increases it could make, [tex]$x$[/tex], and still have a revenue of [tex][tex]$\$[/tex] 1,700$[/tex]?

A. [tex]$-1.25 x^2 - 7.5 x - 1,700 = 0$[/tex]
B. [tex]$-1.25 x^2 + 7.5 x - 1,700 = 0[tex]$[/tex]
C. [tex]$[/tex]-1.25 x^2 - 7.5 x = 0$[/tex]
D. [tex]$-1.25 x^2 + 7.5 x = 0$[/tex]



Answer :

GinnyAnswer
To solve the problem of determining the number of price increases [tex]\( x \)[/tex] the theater company can make while still achieving a revenue of \[tex]$1,700, follow these steps: 1. Identify the current price and sales: - Current ticket price: \$[/tex]8.50
- Current sales: 200 tickets

2. Express the new price and sales with respect to the number of [tex]\( \$0.25 \)[/tex] increases [tex]\( x \)[/tex]:
- New ticket price: [tex]\( 8.50 + 0.25x \)[/tex]
- Decrease in sales: [tex]\( 5x \)[/tex]
- New sales: [tex]\( 200 - 5x \)[/tex]

3. Write the revenue formula with the new price and sales:
Revenue = (New ticket price) [tex]\(\times\)[/tex] (New sales)
= [tex]\((8.50 + 0.25x) \times (200 - 5x)\)[/tex]

4. Set the revenue formula equal to the target revenue of \$1700 to form an equation:
[tex]\[ (8.50 + 0.25x) \times (200 - 5x) = 1700 \][/tex]

5. Expand and simplify the equation:
[tex]\[ (8.50 + 0.25x) \times (200 - 5x) = 1700 \][/tex]
First, expand the left-hand side:
[tex]\[ 8.50 \times 200 + 8.50 \times (-5x) + 0.25x \times 200 + 0.25x \times (-5x) \][/tex]
Simplify each term:
[tex]\[ 1700 - 42.5x + 50x - 1.25x^2 \][/tex]
Combine like terms:
[tex]\[ 1700 + 7.5x - 1.25x^2 \][/tex]
Set the equation to be equal to the target revenue:
[tex]\[ 1700 + 7.5x - 1.25x^2 = 1700 \][/tex]

6. Subtract 1700 from both sides to obtain a standard quadratic equation form:
[tex]\[ 0 = -1.25x^2 + 7.5x \][/tex]

By following these steps, we get the equation:
[tex]\[ -1.25x^2 + 7.5x - 1700 = 0 \][/tex]

Hence, the correct answer is:
[tex]\[ \boxed{B. \ -1.25 x^2 + 7.5 x - 1,700 = 0} \][/tex]

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