Question

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

Accepted Answer

Understand the problem: You are given a piece of wire of fixed length, and you need to divide it into two parts. One part will be bent into a square, and the other part will be bent into a circle. The goal is to optimize a quantity (e.g., minimize or maximize the total area enclosed by the square and the circle). Define variables: Let the total length of the wire be $$ L . $$Let $$ x $$ represent the length of the wire used to form the square, and $$ L - x $$ represent the length of the wire used to form the circle. Express the areas: The perimeter of the square is $$ x $$, so each side of the square is $$ \frac{x}{4} $$, and the area of the square is $$ \left(\frac{x}{4}\right)^2 = \frac{x^2}{16} . $$For the circle, the circumference is $$ L - x $$, so the radius is $$ \frac{L - x}{2\pi} $$, and the area of the circle is $$ \pi \left(\frac{L - x}{2\pi}\right)^2 = \frac{(L - x)^2}{4\pi} . $$Set up the objective function: The total area enclosed by the square and the circle is $$ A(x) = \frac{x^2}{16} + \frac{(L - x)^2}{4\pi} . $$Depending on the problem's goal (e.g., minimize or maximize the total area), this is the function you will optimize. Find the critical points: Differentiate $$ A(x) $$ with respect to $$ x  to $$find $$ A'(x) . $$Solve $$ A'(x) = 0  to $$find the critical points. Then, use the second derivative test or analyze the behavior of $$ A(x)  to $$determine whether each critical point corresponds to a minimum or maximum.

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

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Key Concepts

Perimeter and Area

Geometric Shapes

Optimization