Science:Math Exam Resources/Courses/MATH104/December 2012/Question 03
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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 dollars each. When y bolts are produced, they can be sold for 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.

Solution 

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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 Since nuts and bolts weigh 0.5kg each and we only have 100kg of steel, our constraint becomes Using the constraint, we can express the objective function in terms of just one variable. Solving for gives . Substituting into we get, We will only concern ourselves with values of since we can't sell negative numbers of nuts. To determine the critical points, solve for . Since this is the only critical point of , if we can show that , then has a global maximum at . We have that and so for all x. Therefore, , and has a global maximum at . The corresponding value of bolts is . Therefore, we should sell 75 nuts and 125 bolts to maximize revenue. 