Science:Math Exam Resources/Courses/MATH110/April 2014/Question 04

MATH110 April 2014

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Question 04
A farmer wants to enclose a rectangular pasture of $500$ square metres. Three sides of the pasture will be enclosed with cedar fencing at a cost of $ $10$ per metre. The remaining side is to be enclosed with a stone wall at a cost of $ $15$ per metre. What should the dimensions of the field be to minimize the cost of the enclosure?

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
Draw a diagram to illustrate the setting and label the diagram with variables that might be useful later.

Hint 2
What is the variable that we are trying to optimise?

Hint 3
What equations might be useful in relating the variables we have?

Hint 4
Can we express the variable we are trying to optimise as a function of a single 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. We proceed using the standard template for solving optimisation questions. The exact steps might vary between course and section, but the overall steps will be similar. A diagram. So let's define the rectangular pasture to have length $l$ and width $w$. We want to minimise cost $C$. Total cost of fencing will be $C=10w+10w+10l+15l=20w+25l$. We also need to have the area be $500=lw$. Combining the two equations we get: $C(l)=25l+20\left({\frac {500}{l}}\right)=25l+{\frac {10000}{l}}$. In particular, the domain of the function $C$ is $l\in (0,\infty )$. Differentiate, we get: $C'(l)=25-{\frac {10000}{l^{2}}}$. In the domain, the critical point is the solution of $25l^{2}=10000$ which is $l=\pm 20$. We disregard the negative solution not in the domain. So that means $l=20$. To see that it's a global min, Interval $C'(l)$ INC/DEC $(0,20)$ $C'(10)<0$ DEC $(20,\infty )$ $C'(100)>0$ INC So that means $l=20$ is a local min. Since it is decreasing to the left, $C(20)$ must be lower than all other values to the left. By the same argument, it must be lower than all the points to the right as well. So this is indeed a global min. The lowest cost for the fence can be reached by making the length of $20m$ and width $25m$.

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