A certain band charges $142 per ticket for concerts when 1918 people attend. The band members realize that for every $3 increase in the ticket price 21 fewer people will attend the concert. Find a revenue function R in terms of x where x represents the number of additional $3 increases in the ticket price.

What should the ticket price be for the band to maximize its revenue?
How many people will attend the concert when the revenue is maximized?

revenue = price * #tickets, so

R(x) = (142+3x)(1918-21x)
Now just find the vertex of that parabola, and use that to get the #tickets.

what did you get for the # of tickets because for some reason when I keep entering my answer it keeps telling me its wrong

To find the revenue function, we need to determine the relationship between the number of people attending the concert and the ticket price.

We are given that when 1918 people attend the concert, the ticket price is $142. We also know that for every $3 increase in the ticket price, 21 fewer people will attend the concert.

Let's break down the problem step by step:

1. Calculate the initial ticket price and the corresponding number of people attending:
- Ticket price: $142
- Attendees: 1918

2. Calculate the price increase per ticket:
- This is given as $3 per increase.

3. Calculate the decrease in the number of people attending for every $3 increase in the ticket price:
- The problem states that for every $3 increase, 21 fewer people attend.
- So, the decrease in attendees per $3 increase is -21.

4. Determine the relationship between the number of people attending and the ticket price:
- We can write the number of people attending as a function of the ticket price (x).
- The initial number of people attending is 1918, and for each $3 increase in the ticket price, there will be 21 fewer attendees.
- Therefore, the function for the number of people attending (N) can be expressed as: N(x) = 1918 - 21x

5. Calculate the revenue function:
- The revenue is a product of the number of people attending and the ticket price.
- The revenue function (R) can be expressed as: R(x) = N(x) * ($142 + $3x)
- Substituting the value of N(x) into the revenue function, we get:
- R(x) = (1918 - 21x) * ($142 + $3x)

To find the ticket price that maximizes revenue, we need to find the value of x that maximizes the revenue function R(x).

1. Take the derivative of the revenue function R(x) with respect to x.
2. Set the derivative equal to zero and solve for x to find the critical points.
3. Use the second derivative test to determine whether the critical points correspond to a maximum or minimum.
4. Evaluate the revenue function at the critical points and select the ticket price that results in the maximum revenue.

To find the number of people attending when the revenue is maximized, substitute the value of x (the number of $3 increases in the ticket price) that maximizes revenue into the number of attendees function N(x).