Question
The booster club at Preston Hills High School is selling tickets to a fundraiser for the marching band. The club wants to figure out how much to charge per ticket to make $3,600, which is the total cost for six new trumpets. Based on previous fundraisers, the expression
–
4p+260 can be used to estimate the number of tickets the booster club will sell given the ticket price, p.
Which equation can the booster club use to estimate a ticket price that will raise $3,600 in revenue?
What is the lowest ticket price the booster club can use to make $3,600 in revenue?
–
4p+260 can be used to estimate the number of tickets the booster club will sell given the ticket price, p.
Which equation can the booster club use to estimate a ticket price that will raise $3,600 in revenue?
What is the lowest ticket price the booster club can use to make $3,600 in revenue?
Answers
GPT-4o mini
To determine the ticket price \( p \) that the booster club needs to charge in order to raise $3,600, we can set up an equation based on the relationship between ticket price, number of tickets sold, and revenue.
From the information given, the number of tickets sold can be modeled by the expression:
\[ n(p) = -4p + 260 \]
Where:
- \( n(p) \) is the number of tickets sold
- \( p \) is the ticket price
The revenue \( R \) generated from selling tickets is given by the product of the number of tickets sold and the ticket price:
\[ R = p \times n(p) = p \times (-4p + 260) \]
We want the revenue to equal $3,600, so we can set up the equation:
\[ p \times (-4p + 260) = 3600 \]
Now we can expand and rearrange that equation:
\[ -4p^2 + 260p - 3600 = 0 \]
Next, let's simplify the equation:
1. Multiply through by -1 to make the leading coefficient positive:
\[ 4p^2 - 260p + 3600 = 0 \]
2. To make calculations easier, we can divide the entire equation by 4:
\[ p^2 - 65p + 900 = 0 \]
Now we can apply the quadratic formula, which is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1 \), \( b = -65 \), and \( c = 900 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-65)^2 - 4 \times 1 \times 900 = 4225 - 3600 = 625 \]
Now, use the quadratic formula:
\[ p = \frac{65 \pm \sqrt{625}}{2 \times 1} \]
\[ p = \frac{65 \pm 25}{2} \]
Calculating the two potential values for \( p \):
1. \( p = \frac{90}{2} = 45 \)
2. \( p = \frac{40}{2} = 20 \)
Thus, the ticket prices that could generate the necessary revenue are:
- \( p = 45 \)
- \( p = 20 \)
The lowest ticket price that the booster club can use to make $3,600 in revenue is:
**$20.**
From the information given, the number of tickets sold can be modeled by the expression:
\[ n(p) = -4p + 260 \]
Where:
- \( n(p) \) is the number of tickets sold
- \( p \) is the ticket price
The revenue \( R \) generated from selling tickets is given by the product of the number of tickets sold and the ticket price:
\[ R = p \times n(p) = p \times (-4p + 260) \]
We want the revenue to equal $3,600, so we can set up the equation:
\[ p \times (-4p + 260) = 3600 \]
Now we can expand and rearrange that equation:
\[ -4p^2 + 260p - 3600 = 0 \]
Next, let's simplify the equation:
1. Multiply through by -1 to make the leading coefficient positive:
\[ 4p^2 - 260p + 3600 = 0 \]
2. To make calculations easier, we can divide the entire equation by 4:
\[ p^2 - 65p + 900 = 0 \]
Now we can apply the quadratic formula, which is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1 \), \( b = -65 \), and \( c = 900 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-65)^2 - 4 \times 1 \times 900 = 4225 - 3600 = 625 \]
Now, use the quadratic formula:
\[ p = \frac{65 \pm \sqrt{625}}{2 \times 1} \]
\[ p = \frac{65 \pm 25}{2} \]
Calculating the two potential values for \( p \):
1. \( p = \frac{90}{2} = 45 \)
2. \( p = \frac{40}{2} = 20 \)
Thus, the ticket prices that could generate the necessary revenue are:
- \( p = 45 \)
- \( p = 20 \)
The lowest ticket price that the booster club can use to make $3,600 in revenue is:
**$20.**