A movie theatre seats 200 people. For any particular show, the amount of money, m, the theatre makes is a function of the number of people, n, in attendance. If a ticket costs $12 and the cost to the theatre to run one show is $120, state the domain and range of this function.
domain is clearly [0,200]
m = 12n-120
so, the range is [-120,2280]
The domain of the function would be the possible values for the number of people in attendance, n. Since the movie theatre can seat 200 people, the domain would be all the whole numbers from 0 to 200, inclusive.
The range of the function would be the possible values for the amount of money made, m. The amount of money made is calculated by subtracting the cost of running the show ($120) from the total revenue generated by selling tickets. The revenue is equal to the price per ticket ($12) multiplied by the number of people in attendance (n). Therefore, the range of the function would be all the whole numbers greater than or equal to -120, since it's possible that the theatre could make negative profit if fewer than 10 people attend the show, up to the maximum revenue that could be generated by selling out all 200 seats.
To determine the domain and range of the given function, we need to analyze the constraints and specifications mentioned in the problem.
The number of people, n, in attendance must be a valid input for the function. Since a movie theatre cannot have a negative or fractional number of attendees, the domain of this function will be limited to non-negative integers. Moreover, since the theatre can seat a maximum of 200 people, the largest possible value for n is 200. Therefore, the domain of the function is [0, 200], inclusive.
The amount of money made, m, is calculated based on the number of people, n, and the ticket price. Since a ticket costs $12, the value of m will be at least $12 for each value of n from 0 to 200. Additionally, the theatre has an expense of $120 to run a show, which needs to be subtracted from the total revenue to determine the actual profit. Since the number of people cannot be negative, the minimum revenue the theatre can have is $12 (when no one attends) minus the expense, resulting in -$108 (a loss of $108).
Therefore, the range of this function is [-$108, $2400].