Research results for a concert show that 10,000 people will attend the concert if the price is $20 per tickets. For each dollar increase in price, 1000 fewer people will attend.                                                                                          Revenue = (number of people)(ticket price)                                                                                                                    If R(p) represents the total revenue for a ticket price of p dollars, find R(22).

To solve this problem, let's first understand the relationship between the ticket price $ p $ and the number of people who will attend the concert.

### Step 1: Determine the number of attendees as a function of price
- When the ticket price is $20, the number of people attending is 10,000.
- For every $1 increase in price, 1,000 fewer people will attend.

Let $ p $ be the price of the ticket in dollars. The number of people attending can be represented as:

\[
\text{Number of attendees} = 10,000 - 1,000(p - 20)
\]

This equation accounts for the decrease in the number of attendees as the ticket price increases.

### Step 2: Express revenue as a function of price $ p $
Revenue $ R(p) $ is given by:

\[
R(p) = (\text{Number of attendees}) \times (\text{Ticket price})
\]

Substitute the expression for the number of attendees:

\[
R(p) = \left(10,000 - 1,000(p - 20)\right) \times p
\]

Simplify the expression:

\[
R(p) = (10,000 - 1,000p + 20,000) \times p
\]

\[
R(p) = (30,000 - 1,000p) \times p
\]

\[
R(p) = 30,000p - 1,000p^2
\]

### Step 3: Find $ R(22) $
Now, substitute $ p = 22 $ into the revenue function $ R(p) $:

\[
R(22) = 30,000(22) - 1,000(22)^2
\]

Calculate each term:

\[
R(22) = 660,000 - 1,000 \times 484
\]

\[
R(22) = 660,000 - 484,000
\]

\[
R(22) = 176,000
\]

### Final Answer:
The revenue when the ticket price is $22 is **$176,000**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How would the revenue change if the ticket price was $25?
2. What is the maximum possible revenue?
3. How does the revenue function behave as the ticket price increases?
4. What ticket price results in zero attendees?
5. How does the revenue function change if 500 fewer people attend per dollar increase?

**Tip:** When dealing with quadratic functions, always consider checking for maximum or minimum points using calculus or by completing the square.

Research results for a concert show that 10,000 people will attend the concert if the price is $20 per ticket. For each dollar increase in price, 1000 fewer people will attend. Revenue = (number of people)(ticket price). If R(p) represents the total revenue for a ticket price of p dollars, find R(22).

Explore how to calculate concert revenue using the quadratic function R(p) = 30,000p - 1,000p^2. Understand the relationship between ticket price and attendees, and determine the revenue when the ticket price is $22. Suitable for students in Grades 9-11.

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