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Q8:
Find the cost of polishing a circular table-top of diameter $1.6$ m, if the rate of polishing is ₹ $15/m^2$. (Take $\pi = 3.14$)

Solution :

Given Variables & Initial Setup

We are tasked with finding the total cost of polishing the top surface of a circular table. The following parameters are provided:

  • Diameter of the circular table-top ($d$) = $1.6 \text{ m}$
  • Rate of polishing ($R$) = $₹ 15 \text{ per m}^2$
  • Constant approximation for $\pi$ = $3.14$

Step 1: Determining the Radius of the Circular Table-Top

The standard formula for the area of a circle requires the radius ($r$). The radius is defined as exactly half of the diameter [Per the fundamental geometric properties of a circle].

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

$r = \frac{1.6 \text{ m}}{2} = 0.8 \text{ m}$

d = 1.6 m r = 0.8 m Circular Table-Top

Step 2: Calculating the Area of the Circular Region

Polishing is a surface-level operation, meaning we must calculate the two-dimensional area of the table-top. We apply the area formula for a circle [Derived from the integration of concentric circular rings, $A = \pi r^2$].

$A = \pi r^2$

Substituting the given values into the equation:

$A = 3.14 \times (0.8 \text{ m})^2$

First, square the radius:

$(0.8)^2 = 0.8 \times 0.8 = 0.64 \text{ m}^2$

Next, multiply by $\pi$:

$A = 3.14 \times 0.64 \text{ m}^2$

$A = 2.0096 \text{ m}^2$

The total area to be polished is exactly $2.0096 \text{ m}^2$.

Step 3: Computing the Total Cost of Polishing

The total cost ($C$) is the product of the total area and the rate of polishing per square meter. [Dimensional analysis confirms this operation: $\text{m}^2 \times \frac{₹}{\text{m}^2} = ₹$].

$C = \text{Area} \times \text{Rate}$

$C = 2.0096 \text{ m}^2 \times 15 \frac{₹}{\text{m}^2}$

Performing the multiplication:

$C = 30.144 \text{ ₹}$

In standard currency formatting, values are typically rounded to two decimal places (the nearest paisa). Therefore, $30.144$ rounds to $30.14$.

Final Solution: The total cost of polishing the circular table-top is ₹ 30.14.


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