Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 11 (b)/Statement

You and other students in your small class group decide to buy and manage a hotel with 100 rooms. You want to optimize revenue for the hotel, and try charging different amounts per night. When you charge $\mathrm{\$}150$ per night, you rent out 50 rooms per night. When you charge $\mathrm{\$}140$ per night, you rent out 55 rooms per night. It costs $\mathrm{\$}5$ per day to maintain a room that is not rented, and $\mathrm{\$}25$ per day to maintain a room that is rented.

Write the daily profit function $f(p)$.

(Recall that profit is revenue minus costs. Your answer should only depend on $p$, and no other variables.)

Remember to fully justify your solution.