Science:Math Exam Resources/Courses/MATH104/December 2012/Question 03

MATH104 December 2012

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Question 03
A steel company, ABC steel, manufactures nuts and bolts. When x nuts are produced, they can be sold for $\displaystyle {}-3x+500$ dollars each. When y bolts are produced, they can be sold for $\displaystyle {}-y+300$ dollars each. Assume that nuts and bolts weigh 0.5kg each. How many nuts and how many bolts must be produced to maximize the revenue from 100kg of steel? Justify your answer.

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
Remember, revenue is price times quantity. What variables are the quantities, what are the prices?

Hint 2
What is the constraint? How can we use it to write our objective function (revenue) in terms of one variable?

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. In this optimization problem, we want to maximize our revenue, so our objective function is ${\begin{aligned}R&=({\mbox{number of nuts sold}})({\mbox{price per nut}})\\&\,\,+({\mbox{number of bolts sold}})({\mbox{price per bolt}})\\&=(x)(-3x+500)+(y)(-y+300)\\&=500x+300y-3x^{2}-y^{2}.\end{aligned}}$ Since nuts and bolts weigh 0.5kg each and we only have 100kg of steel, our constraint becomes $\displaystyle 0.5x+0.5y=100.$ Using the constraint, we can express the objective function in terms of just one variable. Solving for $y$ gives $y=200-x$. Substituting into $R$ we get, ${\begin{aligned}R&=500x+300(200-x)-3x^{2}-(200-x)^{2}\\&=20000+600x-4x^{2}\end{aligned}}$ We will only concern ourselves with values of $x>0$ since we can't sell negative numbers of nuts. To determine the critical points, solve $R'(x)=0$ for $x$. ${\begin{aligned}\displaystyle {}R'(x)=600-8x&=0\\x&={\frac {600}{8}}=75.\end{aligned}}$ Since this is the only critical point of $R$, if we can show that $R''(75)<0$, then $R$ has a global maximum at $x=75$. We have that $\displaystyle {}R''(x)=-8$ and so $\displaystyle {}R''(x)<0$ for all x. Therefore, $R''(75)=-8<0$, and $R$ has a global maximum at $x=75$. The corresponding value of bolts is $y=125$. Therefore, we should sell 75 nuts and 125 bolts to maximize revenue.

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