Problem 185 - A Complete Resource Book in Mathematics for JEE Main

Question

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) \(x=2 r\)

Step-by-Step Solution

Answer

Option (C) \((4-\pi)x=\pi r\) is the correct condition for minimizing the sum of areas.

Step 1: Understand the Problem

We have a wire of length 2 units, divided into two parts to form a square and a circle. We need to minimize the sum of the areas of these two geometric shapes.

Step 2: Define Variables and Formulas

Let one part of the wire be used to form a square, giving each side length as \(x\). The perimeter of the square, therefore, is \(4x\). The remaining part forms a circle with radius \(r\), so the circumference of the circle is \(2\pi r\).

Step 3: Set Up the Equation for Perimeter

The entire wire is 2 units, so we have the equation: \(4x + 2\pi r = 2\).

Step 4: Express One Variable in Terms of the Other

From the equation \(4x + 2\pi r = 2\), solve for \(r\): \(2\pi r = 2 - 4x\), so \(r = \frac{2 - 4x}{2\pi}\).

Step 5: Calculate Areas in Terms of Variables

The area of the square is \(x^2\) and the area of the circle is \(\pi r^2\). Substitute \(r = \frac{2 - 4x}{2\pi}\) into the circle's area to express it fully in terms of \(x\).

Step 6: Sum of Areas Function

The sum of the areas is \(f(x) = x^2 + \pi \left(\frac{2 - 4x}{2\pi}\right)^2\). Simplify this to find \(f(x)\) in terms of \(x\).

Step 7: Differentiate the Function

Take the derivative of \(f(x)\) with respect to \(x\) to find \(f'(x)\).

Step 8: Solve for Critical Points

Set \(f'(x) = 0\) and solve for \(x\) to find the points where the area sum is minimal.

Step 9: Check Solutions Against Answers

Plug in the values back into the options to check if they match. Evaluate whether (C) matches the derived condition for the minimum area.

Key Concepts

Geometry

Geometry is the study of shapes, sizes, and the properties of space. In optimization problems like this one, understanding the basic properties of geometric shapes is key. For the square, each side is equal, and the perimeter can be found by adding up the lengths of all sides. This gives us a perimeter formula of \(4x\) for a square with side \(x\).

For the circle, we use the property that relates the circumference to the radius: the circumference \(C = 2\pi r\). Knowing these formulas helps us set up the initial equation concerning how the wire divides and forms these shapes.

• The wire's total length forms the perimeter of the square and the circle combined.
• This fundamental geometrical understanding allows us to express the constraints mathematically.

Calculus

Calculus provides us with tools to find the minima and maxima of functions, which is crucial for solving optimization problems. In this problem, we first express the areas of the square and circle in terms of one variable, \(x\), and derive a function \(f(x)\) representing the sum of these areas.

Differentiation is then used to determine how the function behaves. We take the derivative \(f'(x)\) and find the critical points by setting it to zero. These critical points are potential candidates for where the function reaches its minimum. To ensure it is a minimum, we might also consider the second derivative or analyze values around the critical point.

• Differentiation is powerful because it tells us about the incline or decline of functions.
• The derivative zero implies a potential minimum or maximum.
• Solving \(f'(x) = 0\) gives us the wire division where the area is minimized.

Mathematical Modelling

Mathematical modelling involves representing real-world situations using mathematical concepts and language. In this problem, the wire splitting into a square and a circle is our real-world scenario.

We model this using equations: the balance of wire use forms the equation \(4x + 2\pi r = 2\). We also model each shape's area and combine them into a function to optimize. This represents how the mathematical picture of a problem allows us to apply analysis and find practical solutions like optimal wire cutting.

• Mathematical models simplify real-world problems into manageable equations or functions.
• The model offers a structured way to apply mathematical tools like calculus.
• Connecting geometry with calculus, we translate a physical task into a mathematical framework.