Optimization

Step 1: Define the variables.
Let be the length of the wire used for the square, and be the length of the wire used for the circle. Since the total length of the wire is 28m, we have: Step 2: Express the areas.
- The area of the square is given by: - The circumference of the circle is , so the radius of the circle is . The area of the circle is: Step 3: Define the objective function.
The total area is the sum of the areas of the square and the circle: Step 4: Substitute .
Since , we substitute into the objective function: Step 5: Minimize the objective function.
To minimize the area, we take the derivative of with respect to and set it equal to zero: Simplify: Set to find the critical points: Step 6: Solve for .
Cross-multiply to solve for : Step 7: Conclusion.
Thus, the length of the wire to be used for the square is . The length of the wire to be used for the circle is .