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

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MATH100 C December 2024

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Question 11 (b)

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.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Profit = revenue − cost, where revenue is $x(p)\cdot p$ and cost is $25x(p)+5(100-x(p))$. Substitute your expression for $x(p)$.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

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 $$f(p)=-\frac{1}{2}{p}^2+135p-3000.$$

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