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| Math Central | Quandaries & Queries |
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Question from kobina, a student: 4 ft of a wire is to be used to form a square and a circle. how much of the wire is to be used for the square and how much should be used for the square in order to enclose the maximum total area |
Hi Kobina,
Imagine that you cut the wire into two pieces, one x feet long and then the other will be 4 - x feet long. Use the piece 4 - x feet long to form the square and hence each side of the square will be (4 - x)/4 feet long. Use the other piece to form the circle and hence the circle will have a circumference of x feet, that is x = 2 π r. Solve for r.
Form the function f(x) = (area of the square) + (area of the circle). Use you knowledge of calculus to maximize f(x). You need to take some care when you do this. You know that x is between 0 and 4 and one of the theorems in your text says that a continuous function on a closed interval has a maximum and a minimum. Thus if you assume that 0 ≤ x ≤ 4 then f(x) has a maximum. If 0 < x < 4 then there is no guarantee that f(x) has a maximum value.
Harley
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