Question

a local theater is putting on a play for one night in their high school auditorium. the director creates a function to represent the relationship between their tickets sold, t, and revenue, r(t). what is a reasonable domain and range for this function, and why

Explanation:

Step1: Analyze the domain

The number of tickets sold \(t\) cannot be negative. Also, it is limited by the capacity of the auditorium. Let's assume the auditorium has a maximum capacity \(n\). So, \(t\) can be \(0\leq t\leq n\) where \(t\) is an integer (since you can't sell a fraction of a ticket).

Step2: Analyze the range

Revenue \(R(t)\) is related to the number of tickets sold. If the price per ticket is \(p\), then \(R(t)=p\times t\). When \(t = 0\), \(R(0)=0\). When \(t=n\), \(R(n)=p\times n\). So, \(R(t)\) ranges from \(0\) to \(p\times n\) and \(R(t)\) values are non - negative real numbers (assuming \(p\) is a non - negative real number) and \(R(t)\) will take values such that \(0\leq R(t)\leq pn\)

Answer:

Domain: \(0\leq t\leq n\) (where \(n\) is the capacity of the auditorium and \(t\) is an integer) because the number of tickets sold can't be negative and is limited by the auditorium's capacity. Range: \(0\leq R(t)\leq pn\) (where \(p\) is the price per ticket) because revenue is \(0\) when no tickets are sold and at most \(p\times n\) when all \(n\) tickets are sold.