Q2(b):

Find the area of the following circles, given that: (b) diameter = $49$ m

Given Variables & Initial Setup

We are tasked with determining the area of a two-dimensional circle given its diameter. The primary parameter provided is:

• Diameter ($d$) = $49 \text{ m}$

Step 1: Derivation of the Radius

The radius ($r$) of a circle is defined as the linear distance from the center point to any point on its circumference. [Per Euclidean geometry, the diameter is the longest chord of the circle passing directly through the center, making it exactly twice the length of the radius]. Therefore, we establish the fundamental relationship:

$r = \frac{d}{2}$

Substituting the given diameter into the equation:

$r = \frac{49}{2} \text{ m} = 24.5 \text{ m}$

Geometric Visualization

Below is a scaled geometric representation of the circle, illustrating the relationship between the diameter and the radius.

Step 2: Application of the Area Formula

The area ($A$) of a circle is calculated using the standard formula:

$A = \pi r^2$

[Derived historically via the method of exhaustion by Archimedes, where the area of a circle is proven to be the limit of the areas of inscribed regular polygons as the number of sides approaches infinity].

For this computation, we will utilize the rational approximation $\pi \approx \frac{22}{7}$. This specific approximation is strategically chosen because the radius ($\frac{49}{2}$) contains a numerator that is a multiple of $7$, which will elegantly simplify the subsequent arithmetic.

Step 3: Algebraic Computation

Substitute $r = \frac{49}{2} \text{ m}$ and $\pi = \frac{22}{7}$ into the area formula:

$A = \frac{22}{7} \times \left(\frac{49}{2}\right)^2$

Expand the squared term to prepare for cross-cancellation:

$A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$

Perform cross-cancellation to simplify the fractional expression. Divide $22$ by $2$ to yield $11$, and divide $49$ by $7$ to yield $7$:

$A = 11 \times 7 \times \frac{49}{2}$

Multiply the integer terms:

$A = 77 \times \frac{49}{2}$

Convert the remaining fraction to a decimal for final multiplication ($ \frac{49}{2} = 24.5 $):

$A = 77 \times 24.5$

Execute the final multiplication:

$A = 1886.5 \text{ m}^2$

Final Solution: The area of the circle is $1886.5 \text{ m}^2$.