Last year, 200 people came to see the fall play at Jones High School. The ticket price was $2.00. This year, the drama teacher wants to increase the ticket price. He estimates that for every $0.25 increase, 10 fewer people will come to the play. What ticket price will maximize the income?
income = ticket price * people = (2 + .25 n)(200 - 10 n)
i = 400 + 50 n - 20 n - 2.5 n^2 = -2.5 n^2 + 30 n + 400
the max is on the axis of symmetry ... -b / 2a = -30 / -5 = 6
2 + (.25 * 6) = 3.50
Well, it sounds like the drama teacher wants to find that perfect balance between making more money and having a good audience. Let's see if we can clown around with some numbers to help figure it out!
Last year, with a $2.00 ticket price, 200 people showed up. So, the total income was 200 * $2.00 = $400. Simple enough!
Now, let's consider the drama teacher's estimate. For every $0.25 increase in ticket price, 10 fewer people will come. Let's calculate how many people will show up for different ticket prices:
- If the ticket price is $2.25, then 200 - 10 = 190 people will come.
- If the ticket price is $2.50, then 190 - 10 = 180 people will come.
- If the ticket price is $2.75, then 180 - 10 = 170 people will come.
We can continue this pattern, but let's stop clowning around and use a formula to describe it. If x is the number of $0.25 increases, then the number of people attending the play can be represented by the equation:
Number of people = 200 - 10x
Now, let's look at the income based on the number of people and ticket prices. The income is given by the equation:
Income = (200 - 10x) * (2.00 + 0.25x)
To find the ticket price that maximizes the income, we need to find the value of x that gives us the highest income. Instead of juggling with calculations, let's plot this equation on a graphing calculator or spreadsheet and find the peak point!
Once we find it, we can return to those numbers and determine the ticket price that will maximize the income. It's all a balancing act, after all!
Remember, this is just a clownish way to approach the problem. The actual calculation may require some number crunching and analysis.
To maximize the income from ticket sales, we need to find the ticket price that will generate the highest revenue.
Let's analyze the situation step by step:
1. Start with the given information:
- Last year, 200 people attended the play.
- The ticket price was $2.00.
2. Calculate the total revenue from last year's play:
- Revenue = Number of people * Ticket price
- Revenue = 200 * $2.00 = $400.00
3. Determine the relationship between the ticket price and the number of people attending:
- According to the drama teacher's estimates, for every $0.25 increase in ticket price, 10 fewer people will attend the play.
4. Create a table to analyze different ticket prices and the corresponding number of people attending the play:
| Ticket Price | Number of People |
|--------------|-----------------|
| $2.00 | 200 |
| $2.25 | 190 |
| $2.50 | 180 |
| $2.75 | 170 |
| $3.00 | 160 |
| ... | ... |
Continue this table by increasing the ticket price by $0.25 and subtracting 10 from the previous row's number of people.
5. Calculate the revenue for each ticket price:
- Revenue = Number of people * Ticket price
| Ticket Price | Number of People | Revenue |
|--------------|-----------------|-----------------|
| $2.00 | 200 | $400.00 |
| $2.25 | 190 | $427.50 |
| $2.50 | 180 | $450.00 |
| $2.75 | 170 | $467.50 |
| $3.00 | 160 | $480.00 |
| ... | ... | ... |
6. From the table, we can see that as the ticket price increases, the number of people attending decreases, but the revenue initially increases.
- However, at some point, the decrease in attendance will result in a lower overall revenue.
7. Calculate the revenue for each different ticket price and identify the maximum revenue achieved:
| Ticket Price | Number of People | Revenue |
|--------------|-----------------|-----------------|
| $2.00 | 200 | $400.00 |
| $2.25 | 190 | $427.50 |
| $2.50 | 180 | $450.00 |
| $2.75 | 170 | $467.50 |
| $3.00 | 160 | $480.00 |
| ... | ... | ... |
From the table, we can see that the maximum revenue achieved is $480.00 when the ticket price is $3.00.
Therefore, to maximize the income, the drama teacher should set the ticket price at $3.00.
To find the ticket price that will maximize the income, we need to consider the relationship between the ticket price, the number of people attending, and the income generated.
Let's start by analyzing the relationships mentioned in the question:
1. Ticket Price: We know that the initial ticket price was $2.00, and the teacher wants to increase it by $0.25 for each 10 fewer people.
2. Number of People Attending: We are given that for every $0.25 increase in ticket price, 10 fewer people will come to the play.
3. Income: The income is calculated by multiplying the ticket price by the number of people attending.
Now, let's approach this step by step:
1. Determine the number of people attending for each ticket price:
Starting with the original ticket price of $2.00, we can calculate the number of people attending as follows:
- Ticket price: $2.00
- Number of people attending: 200
For each $0.25 increase in ticket price, there will be 10 fewer people attending. This can be represented as:
- Ticket price increase: $0.25
- Decrease in the number of people attending: 10
Using this information, we can calculate the number of people attending for each ticket price increment:
- Ticket price: $2.00 (no increase)
- Number of people attending: 200
- Ticket price: $2.25 (increase by $0.25)
- Number of people attending: 200 - 10 = 190
- Ticket price: $2.50 (increase by $0.25)
- Number of people attending: 190 - 10 = 180
- Continue this pattern until you reach a point where the number of people attending becomes zero or negative.
2. Calculate the income for each ticket price:
The income is calculated by multiplying the ticket price by the number of people attending:
- For $2.00, income = $2.00 x 200 = $400.00
- For $2.25, income = $2.25 x 190 = $427.50
- For $2.50, income = $2.50 x 180 = $450.00
Continue calculating the income for each ticket price increment.
3. Identify the ticket price that maximizes the income:
Compare the income obtained for each ticket price and identify the highest value. The ticket price associated with this income value will be the one that maximizes the income.
In this case, we can see that the income is highest when the ticket price is $2.50, generating $450.00 in revenue. Therefore, increasing the ticket price to $2.50 will maximize the income from the fall play.