The problem in the image involves optimizing the use of a wire of 16 cm, which is divided into two parts. One part is used to form a square, and the other is used to form a circle. The goal is to determine the sizes of these parts to:

1. Minimize the total area of the square and circle.
2. Maximize the total area of the square and circle.

Let me break it down into a mathematical model.

Step 1: Define variables

• Let $x$ be the length of the wire used for the square.
• The remaining wire, $16 - x$, is used for the circle.

Step 2: Relationships

• Perimeter of the square: $x$ (so each side of the square is $\frac{x}{4}$).
• Perimeter of the circle: $16 - x$ (so the radius of the circle is $\frac{16 - x}{2\pi}$).

Areas:

1. Area of the square: $\left(\frac{x}{4}\right)^2 = \frac{x^2}{16}$.
2. Area of the circle: $\pi \left(\frac{16 - x}{2\pi}\right)^2 = \frac{(16 - x)^2}{4\pi}$.

Step 3: Total area

The total area is: $A(x) = \frac{x^2}{16} + \frac{(16 - x)^2}{4\pi}.$

Step 4: Optimization

a. Minimize total area

To find the value of $x$ that minimizes $A(x)$, we take the derivative $A'(x)$, set it to 0, and solve for $x$.

b. Maximize total area

Similarly, we solve for $x$ that maximizes $A(x)$.

Let me compute these solutions mathematically.import sympy as sp

Define variables

x = sp.symbols('x', positive=True) pi = sp.pi

Total area equation

A = (x**2 / 16) + ((16 - x)**2 / (4 * pi))

Derivative of the area

A_prime = sp.diff(A, x)

Solve for critical points

critical_points = sp.solve(A_prime, x)

Evaluate the second derivative to determine minima or maxima

A_double_prime = sp.diff(A_prime, x) results = [] for point in critical_points: curvature = A_double_prime.subs(x, point) area_value = A.subs(x, point) results.append((point, curvature, area_value))

Add endpoints x = 0 and x = 16 for comparison

endpoints = [(0, A.subs(x, 0)), (16, A.subs(x, 16))]

results, endpoints([(64/(pi + 4), 1/8 + 1/(2pi), (16 - 64/(pi + 4))**2/(4pi) + 256/(pi + 4)**2)], [(0, 64/pi), (16, 16)])([(64/(pi + 4), 1/8 + 1/(2pi), (16 - 64/(pi + 4))**2/(4pi) + 256/(pi + 4)**2)], [(0, 64/pi), (16, 16)])success