Math CentralQuandaries & Queries


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.


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