Q13:

A circular flower bed is surrounded by a path $4$ m wide. The diameter of the flower bed is $66$ m. What is the area of this path? ($\pi = 3.14$)

Given Variables & Initial Setup

We are analyzing a geometric system consisting of two concentric circles: an inner circular flower bed and an outer boundary that includes a surrounding path. The fundamental parameters provided are:

• Diameter of the inner flower bed, $d = 66 \text{ m}$
• Width of the surrounding path, $w = 4 \text{ m}$
• Constant for Pi, $\pi = 3.14$

Step 1: Determining the Inner and Outer Radii

To compute the area of circular regions, we must first determine their respective radii.

1. Inner Radius ($r$):
[Per the fundamental property of a circle, the radius is exactly half of the diameter.]
$r = \frac{d}{2}$
$r = \frac{66 \text{ m}}{2} = 33 \text{ m}$

2. Outer Radius ($R$):
[The outer circle encompasses both the inner flower bed and the uniform path. Therefore, the outer radius is the sum of the inner radius and the path width.]
$R = r + w$
$R = 33 \text{ m} + 4 \text{ m} = 37 \text{ m}$

Step 2: High-Precision Geometric Representation

Below is a scaled, mathematically accurate representation of the concentric circles forming the annulus (the path). (Scale: 1 meter = 5 SVG units)

Step 3: Formulating the Area of the Path (Annulus)

[Per the geometric definition of an annulus, the area of the region bounded by two concentric circles is the difference between the area of the outer circle and the area of the inner circle.]

Let $A$ be the area of the path.
$A = \text{Area of Outer Circle} - \text{Area of Inner Circle}$
$A = \pi R^2 - \pi r^2$

Factoring out the common constant $\pi$ yields:
$A = \pi (R^2 - r^2)$

Step 4: Algebraic Computation

Substitute the established values ($R = 37$, $r = 33$, $\pi = 3.14$) into the area formula:

$A = 3.14 \times (37^2 - 33^2)$

[Applying the algebraic identity for the difference of two squares: $a^2 - b^2 = (a - b)(a + b)$ to streamline the arithmetic without calculating large squares.]

$A = 3.14 \times [(37 - 33)(37 + 33)]$

Calculate the terms inside the brackets:
$37 - 33 = 4$
$37 + 33 = 70$

Substitute these back into the equation:
$A = 3.14 \times (4 \times 70)$
$A = 3.14 \times 280$

Perform the final multiplication:
$A = 3.14 \times 280 = 879.2$

Final Solution: The area of the path surrounding the flower bed is $879.2 \text{ m}^2$.