1. ## Optimization problem

An amateur theater company offers a group discount for its evening performances. For every person in a group of 30 or more, the ticket price decreases by $1.50 from the regular price of$24.00. Determine the size of group that maximizes revenue.

How do I go about solving this?

2. The question does not make sense. If you need to be 30 in a group in order to receive a discount, then any group size below 30 is going to maximize the revenue as long as the theater gets filled.

3. Originally Posted by Mondreus
The question does not make sense. If you need to be 30 in a group in order to receive a discount, then any group size below 30 is going to maximize the revenue as long as the theater gets filled.
Well, considering the answer that they give you on the book is '30' then what you are saying makes sense. Revenue starts decreasing when they sell more than 30 tickets so 30 is the critical point in which revenue is maximized.

Could you help me prove this mathematically though?

4. If this is how the question was originally phrased, then I'd still say that it does not make sense to be honest.

If the ticket price decreases with $1.50 for every person in a group of 30 or above, then the ticket price would be less than zero for every group of 30 or above:$\displaystyle 24-1.5\cdot 30 = -21\$

You can express it mathematically with the help of a shifted Heaviside step function, but I seriously doubt that the original question was phrased like this.