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Q6:
Saima wants to put a lace on the edge of a circular table cover of diameter $1.5$ m. Find the length of the lace required and also find its cost if one meter of the lace costs ₹ $15$. (Take $\pi = 3.14$)

Solution :

Step 1: Given Variables & Initial Setup

To determine the length and cost of the lace required for the circular table cover, we first extract the given geometric and financial parameters:

  • Diameter of the circular table cover ($d$): $1.5 \text{ m}$
  • Radius of the circular table cover ($r$): $r = \frac{d}{2} = \frac{1.5}{2} = 0.75 \text{ m}$
  • Cost of lace per meter: ₹ $15$
  • Value of Pi ($\pi$): $3.14$ [As explicitly specified in the problem constraints]

Step 2: Geometric Visualization

The lace is to be attached to the outer edge of the circular table cover. In Euclidean geometry, the continuous line forming the boundary of a closed geometric figure is its perimeter. For a circle, this boundary is defined as the circumference.

d = 1.5 m r = 0.75 m Lace Boundary (Circumference)

Step 3: Calculating the Length of the Lace Required

The total length of the lace required is mathematically equivalent to the circumference of the circle. [Per the geometric definition of a circle's perimeter, $C = \pi d$ or $C = 2\pi r$].

Using the diameter formula for circumference:

$C = \pi \times d$

Substituting the given values into the equation:

$C = 3.14 \times 1.5 \text{ m}$

Performing the arithmetic multiplication:

$C = 4.71 \text{ m}$

Therefore, the exact length of the lace required to cover the edge of the table is $4.71 \text{ m}$.

Step 4: Calculating the Total Cost of the Lace

The total financial cost is determined by taking the product of the total length of the lace and the unit cost per meter. [By the principle of linear proportionality in unit pricing].

$\text{Total Cost} = \text{Total Length of Lace} \times \text{Cost per meter}$

Substituting the calculated length and the given rate:

$\text{Total Cost} = 4.71 \text{ m} \times 15 \text{ ₹/m}$

Breaking down the multiplication for precision:

  • $4.71 \times 10 = 47.10$
  • $4.71 \times 5 = 23.55$
  • $47.10 + 23.55 = 70.65$

$\text{Total Cost} = 70.65 \text{ ₹}$


Final Solution: The total length of the lace required to cover the edge of the circular table is $4.71 \text{ m}$, and the total cost to purchase this lace is ₹ $70.65$.


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