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Q5:

From a circular sheet of radius $4$ cm, a circle of radius $3$ cm is removed. Find the area of the remaining sheet. (Take $\pi = 3.14$)

Solution :

Given Variables & Initial Setup

We are analyzing a composite two-dimensional geometric system where a smaller circle is concentrically removed from a larger circular sheet. The parameters defining this system are:

  • Radius of the outer circular sheet, $R = 4 \text{ cm}$
  • Radius of the inner removed circle, $r = 3 \text{ cm}$
  • Approximation for Pi, $\pi = 3.14$

Geometric Visualization

The figure below illustrates the concentric nature of the two circles. The region between the outer boundary and the inner boundary represents the remaining sheet, geometrically defined as an annulus.

O r = 3 cm R = 4 cm Removed Region Remaining Sheet (Annulus)

Step 1: Formulating the Area Equation

To find the area of the remaining sheet, we must subtract the area of the inner circle from the area of the outer circular sheet. Let $A$ represent the area of the remaining sheet.

$A = \text{Area of Outer Circle} - \text{Area of Inner Circle}$

$A = \pi R^2 - \pi r^2$ [Per the standard area formula for a circle, $Area = \pi \times \text{radius}^2$]

To optimize the calculation, we factor out the common constant $\pi$:

$A = \pi (R^2 - r^2)$ [By the distributive property of multiplication over subtraction]

Step 2: Algebraic Substitution

Substitute the given numerical values into the factored equation:

$A = 3.14 \times (4^2 - 3^2)$

Step 3: Evaluating the Expression

First, resolve the exponents inside the parentheses:

  • $4^2 = 4 \times 4 = 16$
  • $3^2 = 3 \times 3 = 9$

Substitute these squared values back into the equation:

$A = 3.14 \times (16 - 9)$

Perform the subtraction operation:

$A = 3.14 \times 7$

Step 4: Final Computation

Multiply the remaining terms to find the final area:

$A = 21.98$

Since the radii were provided in centimeters ($\text{cm}$), the resulting area must be expressed in square centimeters ($\text{cm}^2$) [Per the dimensional analysis of area, $L \times L = L^2$].

Final Solution: The area of the remaining sheet is $21.98 \text{ cm}^2$.


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