The volume of a cubical solid is ` 10368 cm ^(3) ` If its diamensions are in the ratio 3 : 2: 1 , find the cost of polishing its total surface at the rate of ` 2.50 " per " m^(2)`

Answer

To solve the problem step by step, we will follow the given information and perform the necessary calculations.

### Step 1: Understand the Volume of the Solid
The volume of the cubical solid is given as \( 10368 \, \text{cm}^3 \).

### Step 2: Set Up the Dimensions
The dimensions of the solid are in the ratio \( 3:2:1 \). Let the dimensions be:
- Length \( = 3k \)
- Width \( = 2k \)
- Height \( = k \)

### Step 3: Write the Volume Formula
The volume \( V \) of a cuboid is given by:
\[ V = \text{Length} \times \text{Width} \times \text{Height} = (3k) \times (2k) \times (k) = 6k^3 \]
Given that the volume is \( 10368 \, \text{cm}^3 \), we can set up the equation:
\[ 6k^3 = 10368 \]

### Step 4: Solve for \( k^3 \)
To find \( k^3 \), divide both sides by 6:
\[ k^3 = \frac{10368}{6} = 1728 \]

### Step 5: Find \( k \)
Now, we need to find \( k \) by taking the cube root of \( 1728 \):
\[ k = \sqrt[3]{1728} = 12 \]

### Step 6: Calculate the Dimensions
Now that we have \( k \), we can find the actual dimensions:
- Length \( = 3k = 3 \times 12 = 36 \, \text{cm} \)
- Width \( = 2k = 2 \times 12 = 24 \, \text{cm} \)
- Height \( = k = 12 \, \text{cm} \)

### Step 7: Calculate the Total Surface Area
The total surface area \( A \) of a cuboid is given by:
\[ A = 2(ab + bc + ac) \]
Substituting the dimensions:
\[ A = 2((36 \times 24) + (24 \times 12) + (12 \times 36)) \]

Calculating each term:
- \( 36 \times 24 = 864 \)
- \( 24 \times 12 = 288 \)
- \( 12 \times 36 = 432 \)

Now sum them:
\[ A = 2(864 + 288 + 432) = 2(1584) = 3168 \, \text{cm}^2 \]

### Step 8: Convert to Square Meters
Since the cost is given per square meter, we convert square centimeters to square meters:
\[ 3168 \, \text{cm}^2 = \frac{3168}{10000} \, \text{m}^2 = 0.3168 \, \text{m}^2 \]

### Step 9: Calculate the Cost of Polishing
The cost of polishing is given at the rate of \( 2.50 \) rupees per square meter:
\[ \text{Cost} = \text{Area} \times \text{Rate} = 0.3168 \, \text{m}^2 \times 2.50 \, \text{rupees/m}^2 \]
Calculating the cost:
\[ \text{Cost} = 0.3168 \times 2.50 = 0.792 \, \text{rupees} \]

### Final Answer
The cost of polishing the total surface area is approximately \( 0.79 \, \text{rupees} \).

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