Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 11 (b)/Solution 1

Revenue is given by: $x(p)\cdot p.$

The cost consists of $25$ dollars per rented room and $5$ dollars per non-rented room. Since there are 100 rooms, the cost is $$\text{Cost}=25x(p)+5(100-x(p)).$$

Thus, profit is $$f(p)=x(p)\cdot p-25x(p)-5(100-x(p)).$$

Simplifying, $$f(p)=x(p)(p-25+5)-500=x(p)(p-20)-500.$$

Substitute $x(p)=-\frac{1}{2}p+125$: $$f(p)=(-\frac{1}{2}p+125)(p-20)-500.$$

Expanding, $$f(p)=-\frac{1}{2}{p}^2+10p+125p-2500-500=-\frac{1}{2}{p}^2+135p-3000.$$

Thus, the daily profit function is $$f(p)=-\frac{1}{2}{p}^2+135p-3000.$$