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Q4:

A gardener wants to fence a circular garden of diameter $21$ m. Find the length of the rope he needs to purchase, if he makes $2$ rounds of fence. Also find the cost of the rope, if it costs ₹ $4$ per meter. (Take $\pi = \frac{22}{7}$)

Solution :

Step 1: Given Variables & Initial Setup

Let us define the geometric and financial parameters provided for the circular garden:

  • Diameter of the circular garden, $d = 21 \text{ m}$
  • Radius of the circular garden, $r = \frac{d}{2} = \frac{21}{2} \text{ m} = 10.5 \text{ m}$
  • Number of rounds of fencing required, $n = 2$
  • Cost of the rope per meter, $C_{\text{unit}} = \text{₹ } 4 \text{ per m}$
  • Approximation for Pi, $\pi = \frac{22}{7}$

Step 2: Geometric Visualization

The following diagram illustrates the circular garden from a top-down perspective, including the diameter and the two concentric rounds of rope fencing required.

d = 21 m 2 Rounds of Rope Fence r = 10.5 m

Step 3: Calculating the Circumference of the Garden

The length of rope required for a single round of fencing is exactly equal to the perimeter (circumference) of the circular garden. [Per the geometric definition of a circle's circumference, the boundary length is a function of its diameter].

The formula for the circumference $C$ is given by:

$C = \pi d$

Substituting the given values into the equation:

$C = \left(\frac{22}{7}\right) \times 21 \text{ m}$

By simplifying the fraction [dividing $21$ by $7$ yields $3$]:

$C = 22 \times 3 \text{ m} = 66 \text{ m}$

Thus, one complete round around the garden requires $66 \text{ m}$ of rope.

Step 4: Determining the Total Length of Rope Required

Since the gardener intends to make $2$ complete rounds of the fence, the total length of the rope ($L_{\text{total}}$) is the circumference multiplied by the number of rounds. [Total Length = $n \times C$].

$L_{\text{total}} = 2 \times 66 \text{ m}$

$L_{\text{total}} = 132 \text{ m}$

Step 5: Calculating the Total Cost of the Rope

The total financial cost is the product of the total length of the rope and the unit cost per meter. [Total Cost = $L_{\text{total}} \times C_{\text{unit}}$].

$\text{Total Cost} = 132 \text{ m} \times \text{₹ } 4 / \text{m}$

$\text{Total Cost} = \text{₹ } 528$

Final Solution: The total length of the rope required to fence the garden with 2 rounds is $132 \text{ m}$, and the total cost of purchasing the rope is ₹ $528$.


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