A wire of length a is cut into two parts which are bent,...

A wire of length $a$ is cut into two parts which are bent, respectively, in the form of a square and a circle. The least value of the sum of the areas so formed is
(A) $\frac{a^{2}}{\pi+4}$
(B) $\frac{\mathrm{a}}{\pi+4}$
(C) $\frac{a}{4(\pi+4)}$
(D) $\frac{a^{2}}{4(\pi+4)}$

Key Concepts

Differentiation and Critical Point Analysis

Calculus techniques, particularly differentiation, are used to locate the critical points of a function where its derivative equals zero. These critical points are analyzed to determine if they represent a minimum or a maximum in the context of the problem.

Constraint Handling

When dealing with optimization problems, constraints (such as a fixed total length or perimeter) are used to reduce the number of independent variables. This substitution simplifies the objective function so that standard calculus techniques can be applied to find the optimal solution.

Geometric Area Formulas

Knowledge of geometric formulas, such as the area formulas for a square (side squared) and a circle (pi times the square of the radius), is essential. These formulas help in expressing the area in terms of the dimensions related to the fixed resource, like the total length of the wire.

Optimization

Optimization involves finding the best solution from a set of available alternatives, often by minimizing or maximizing a function. In calculus, this is done by expressing a quantity as a function of variables, then using techniques such as differentiation to find critical points that provide the desired minimum or maximum.

Objective Function Formation

In many problems, including problems involving geometric shapes, the first step is to express the quantity to be optimized, such as total area, as a function of one or more independent variables. This requires relating the quantities through known formulas and imposed constraints.

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