Science:Math Exam Resources/Courses/MATH110/April 2017/Question 10
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Question 10 |
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Show that among all the rectangles with a given area (where is a positive constant), the rectangle with the smallest perimeter is a square. |
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 |
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Let denote the longer side of the rectangle, and be the shorter side. Then by the definition of the area of rectangle, we know that . Then try to find a function of the perimeter and minimise it. |
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.
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Solution |
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Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, we know that the perimeter is In order to find the minimal value of , we differentiate to find its critical points. implies ( rejects the negative value). Thus Note that this gives the minimal perimeter as Thus, the rectangle with the smallest perimeter is a square. |