Science:Math Exam Resources/Courses/MATH104/December 2014/Question 04

MATH104 December 2014

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Question 04
You are planning a city tour for a group of 100 tourists. If you can sell $x$ bus tour tickets, you can offer them for $\$(30-x/4)$ each. If you can sell $y$ train tour tickets, you can offer them for $\$(70-y/2)$ each. How many bus tickets, and how many train tickets should you sell to the tourists in order to maximize revenue (you can only sell one type of ticket to each passenger).

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
If you sell $x$ bus tour tickets, and there are a 100 passengers, what is $y$?

Hint 2
What does maximizing revenue mean mathematically, in terms of derivatives?

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We know that if we sell $x$ bus tour tickets, we sell $y=100-x$ train tour tickets. Let us now make the revenue function: ${\begin{aligned}R&=x(30-{\frac {x}{4}})+y(70-{\frac {y}{2}})\\&=x(30-{\frac {x}{4}})+(100-x)(70-{\frac {100-x}{2}})\\&=30x-{\frac {x^{2}}{4}}+70(100-x)-{\frac {(100-x)^{2}}{2}}\\\end{aligned}}$ The maximum of $R$ can either be at the boundary of the interval or a local maximum. For $x=0$ we find $R(0)=2000$ and for $x=100$ we obtain $R(100)=2500$. To find the critical point we take the derivative of $R$ with respect to $x$. Being careful with the chain rule in the last summand we obtain ${\begin{aligned}R'(x)&=30-{\frac {x}{2}}-70-(100-x)(-1)\\&=60-{\frac {3x}{2}}\end{aligned}}$ Now we want to maximize the revenue with respect to the number of bus tickets we sell; hence we let $R'(x)=0$ and determine $x$. We obtain that $x=40$ and $R(40)=3200$. This means that if we sell 40 bus tour tickets and 60 train tour tickets, we maximize the revenue.

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