A wire of length 2 units is cur into two parts which are...

A wire of length 2 units is cur into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=\mathrm{r}$ units. If the sum of the areas of the square and the circle so formed is minimum, then:(A) $2 x=r$ (B) $2 x=(\pi+4) r$ (C) $(4-\pi) x=\pi r$ (D) $x=2 r$

A wire of length 2 units is cur into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=\mathrm{r}$ units. If the sum of the areas of the square and the circle so formed is minimum, then:(A) $2 x=r$
(B) $2 x=(\pi+4) r$
(C) $(4-\pi) x=\pi r$
(D) $x=2 r$

Key Concepts

Optimization

This problem showcases the need to minimize an objective function (the total area formed by two different shapes) subject to a fixed resource (the total wire length). Optimization techniques involve expressing the objective function in terms of one or more variables and then finding the values at which the function is minimized or maximized.

Calculus and Differentiation

Calculus is a key tool in optimization problems, particularly through the use of derivatives. By taking the derivative of the objective function with respect to the variable and setting it to zero, one can determine the critical points where the function is minimized or maximized.

Constraint Management

In many optimization problems, the variables are bound by a constraint (in this case, the total length of the wire). Formulating this constraint mathematically allows one to express one variable in terms of the others, effectively reducing the complexity of the problem by lowering the number of independent variables.

Geometric Relationships

Understanding the geometric properties of shapes plays an essential role. For example, the area and perimeter of a square are directly related to its side length, and similarly for a circle, its area and circumference are related to its radius. These relationships are critical in setting up the proper equations that form the basis of the optimization problem.

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