The Green Tomato, a new restaurant, is having its grand opening this weekend. To help generate buzz for the event, the owners are raffling off a weekend getaway to the Catskill Mountains. They want to know how much to charge per raffle ticket to make $1,260, which is the total cost of the trip. Based on responses to an online survey, the expression –5p+160 can be used to estimate the number of tickets the owners will sell based on the ticket price, p.What two ticket prices will bring in exactly $1,260 in revenue?
Question
The Green Tomato, a new restaurant, is having its grand opening this weekend. To help generate buzz for the event, the owners are raffling off a weekend getaway to the Catskill Mountains. They want to know how much to charge per raffle ticket to make 1,260 in revenue?
Solution
To find the ticket prices that will bring in exactly 1,260.
The equation is: p(-5p + 160) = 1260
This simplifies to: -5p^2 + 160p = 1260
Rearrange the equation to: 5p^2 - 160p + 1260 = 0
Divide the entire equation by 5 to simplify: p^2 - 32p + 252 = 0
This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for p using the quadratic formula, which is p = [-b ± sqrt(b^2 - 4ac)] / (2a).
In this case, a = 1, b = -32, and c = 252.
Substitute these values into the quadratic formula:
p = [32 ± sqrt((-32)^2 - 41252)] / (2*1) p = [32 ± sqrt(1024 - 1008)] / 2 p = [32 ± sqrt(16)] / 2 p = [32 ± 4] / 2
So, the two possible values for p (the ticket price) are:
p = (32 + 4) / 2 = 18 p = (32 - 4) / 2 = 14
Therefore, the two ticket prices that will bring in exactly 14 and $18.
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