Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 11 (a)/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.

Assume that the demand $x$ for rooms is linear in price $p$. Determine the function $x(p).$

Remember to fully justify your solution.