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.
R = (1918 - 21 x) (142 + 3 x)
two other questions for this problem is:
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?
To find the revenue function R in terms of x, we need to analyze the given information and calculate the revenue generated at each ticket price level.
Let's break down the given information step by step:
1. The initial ticket price is $142, and the number of attendees is 1918 people.
- Revenue at this ticket price = Ticket price * Number of attendees = $142 * 1918 = $<<142*1918=272756>>272,756.
2. For every $3 increase in the ticket price, 21 fewer people will attend the concert.
- This means that for each additional x ($3) increase in the ticket price, the number of attendees will decrease by 21 * x.
Now, let's define the revenue function R in terms of x:
a) Calculate the number of attendees based on the additional $3 increases in the ticket price:
- Initial number of attendees = 1918 - (21 * 0) = 1918.
- Number of attendees after x additional $3 increases = 1918 - (21 * x).
b) Calculate the ticket price after x additional $3 increases:
- Initial ticket price = $142.
- Ticket price after x additional $3 increases = $142 + ($3 * x) = $142 + 3x.
c) Calculate the revenue R by multiplying the ticket price and the number of attendees:
- Revenue = (Ticket price after x additional $3 increases) * (Number of attendees after x additional $3 increases).
- R(x) = ($142 + 3x) * (1918 - 21x).
Therefore, the revenue function R in terms of x is:
R(x) = ($142 + 3x) * (1918 - 21x).