Question
year, tickets sold for $15 each and 600 people attended.
This year, the student council has determined that, for every $1.00
increase in price, there will be 30 fewer students attending. Be sure to define
This year, the student council has determined that, for every $1.00
increase in price, there will be 30 fewer students attending. Be sure to define
Answers
GPT-4o mini
To analyze the situation regarding ticket sales and attendance, we can define some variables and formulate a model based on the information provided.
### Definitions:
1. **Base Ticket Price (P₀)**: The original ticket price, which is $15.
2. **Current Attendance (A₀)**: The original number of attendees, which is 600.
3. **Increase in Ticket Price (x)**: The number of $1 increases in the ticket price.
4. **New Ticket Price (P)**: The adjusted ticket price after x increases:
\[
P = P₀ + x = 15 + x
\]
5. **New Attendance (A)**: The adjusted number of attendees, which decreases by 30 for each $1 increase in price:
\[
A = A₀ - 30x = 600 - 30x
\]
### Problem Formulation:
We're looking for a relationship between the new ticket price and the new attendance. The equations derived above can help express the total revenue generated from ticket sales.
### Revenue Calculation:
The total revenue (R) from ticket sales after the price increase and corresponding drop in attendance is given by:
\[
R = P \cdot A = (15 + x) \cdot (600 - 30x)
\]
### Expanding the Revenue Equation:
By expanding the revenue equation:
\[
R = (15 + x)(600 - 30x)
\]
\[
R = 15 \cdot 600 - 15 \cdot 30x + 600x - 30x^2
\]
\[
R = 9000 - 450x + 600x - 30x^2
\]
\[
R = 9000 + 150x - 30x^2
\]
### Summary:
To summarize, you have a revenue function dependent on the number of $1 increases in ticket price (x). As x increases, the ticket price goes up while attendance goes down, allowing you to model and potentially optimize total revenue generated from the event.
The total revenue function for varying ticket prices and attendance numbers can be analyzed further to determine the optimal price point that maximizes revenue.
### Definitions:
1. **Base Ticket Price (P₀)**: The original ticket price, which is $15.
2. **Current Attendance (A₀)**: The original number of attendees, which is 600.
3. **Increase in Ticket Price (x)**: The number of $1 increases in the ticket price.
4. **New Ticket Price (P)**: The adjusted ticket price after x increases:
\[
P = P₀ + x = 15 + x
\]
5. **New Attendance (A)**: The adjusted number of attendees, which decreases by 30 for each $1 increase in price:
\[
A = A₀ - 30x = 600 - 30x
\]
### Problem Formulation:
We're looking for a relationship between the new ticket price and the new attendance. The equations derived above can help express the total revenue generated from ticket sales.
### Revenue Calculation:
The total revenue (R) from ticket sales after the price increase and corresponding drop in attendance is given by:
\[
R = P \cdot A = (15 + x) \cdot (600 - 30x)
\]
### Expanding the Revenue Equation:
By expanding the revenue equation:
\[
R = (15 + x)(600 - 30x)
\]
\[
R = 15 \cdot 600 - 15 \cdot 30x + 600x - 30x^2
\]
\[
R = 9000 - 450x + 600x - 30x^2
\]
\[
R = 9000 + 150x - 30x^2
\]
### Summary:
To summarize, you have a revenue function dependent on the number of $1 increases in ticket price (x). As x increases, the ticket price goes up while attendance goes down, allowing you to model and potentially optimize total revenue generated from the event.
The total revenue function for varying ticket prices and attendance numbers can be analyzed further to determine the optimal price point that maximizes revenue.