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Q2(a):
Find the area of the following circles, given that: (a) radius = $14$ mm (Take $\pi = \frac{22}{7}$)

Solution :

Given Variables & Initial Setup

We are given the following geometric parameters for a circle:

  • Radius ($r$) = $14 \text{ mm}$
  • Mathematical constant ($\pi$) = $\frac{22}{7}$

Theoretical Foundation & Formula

The area ($A$) of a circle represents the total two-dimensional space enclosed within its boundary (circumference). [Per the geometric principles of Euclidean space], the area of a circle is directly proportional to the square of its radius. The formula is given by:

$A = \pi r^2$

O r = 14 mm

Step 1: Substituting the Values into the Equation

To find the area, we substitute the given radius $r = 14 \text{ mm}$ and the fractional approximation of $\pi = \frac{22}{7}$ into the standard area formula:

$A = \left(\frac{22}{7}\right) \times (14 \text{ mm})^2$

Step 2: Algebraic Manipulation and Simplification

First, expand the squared term to separate the numerical values from the units:

$A = \frac{22}{7} \times (14 \times 14) \text{ mm}^2$

$A = \frac{22}{7} \times 196 \text{ mm}^2$

To optimize the calculation without dealing with large numbers, we can simplify the expression by dividing one of the $14$ factors by the denominator $7$ [By the fundamental property of fractions and associative multiplication]:

$A = 22 \times \left(\frac{14}{7}\right) \times 14 \text{ mm}^2$

$A = 22 \times 2 \times 14 \text{ mm}^2$

Now, proceed with sequential multiplication:

$A = (22 \times 2) \times 14 \text{ mm}^2$

$A = 44 \times 14 \text{ mm}^2$

Perform the final multiplication step:

$A = 44 \times (10 + 4) \text{ mm}^2$

$A = 440 + 176 \text{ mm}^2$

$A = 616 \text{ mm}^2$

Step 3: Dimensional Analysis

The radius is given in millimeters ($\text{mm}$). When the radius is squared in the formula ($r^2$), the unit is also squared ($\text{mm} \times \text{mm} = \text{mm}^2$). This confirms that our final unit correctly represents a two-dimensional area.

Final Solution: The area of the circle is $616 \text{ mm}^2$.


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