Final answer:
The problem of splitting a wire into two pieces to form a circle and a square, maximizing or minimizing the total enclosed area, involves using calculus to derive expressions for the areas and then finding the extremum values.
Step-by-step explanation:
The student's question about cutting a wire into two different shapes to either maximize or minimize the total enclosed area falls within the subject of Mathematics. This problem can be approached using calculus by setting up an equation to relate the lengths of the wire segments to the areas of the shapes they form.
To find the maximum total area, we establish the lengths of wire used for the circle and square, express the areas of the circle and the square in terms of these lengths, and then use the method of derivatives to find the maximum area. For the minimum total area, we repeat the same process while looking for the minimum point rather than the maximum.
We are given a total wire length of 3 meters, and two shapes are formed: a circle and a square. The circumference of the circle (C) is 2πr and the perimeter of the square (P) is 4s. The total wire length is the sum of these two, so 3 = 2πr + 4s.
The enclosed areas are πr2 for the circle and s2 for the square. To find the extrema, we have to set up an equation for the total area A = πr2 + s2 and then find the derivative of A with respect to one of the variables, r or s. By making one of the variables the subject in 3 = 2πr + 4s and substituting it into A, we can then find the derivative and solve for the maximum and minimum cases.