The cost of a precious stone varies as the cube of its weight. The stone broke into 3 pieces whose weights are in the ratio ` 1 : 2 : 3`. As a result its cost reduces . If the cost of the unbroken stone is Rs. 96, 336, then find the loss incurred due to breakage.

Options

A. ` Rs. 80,280`
B. ` Rs. 16,056`
C. ` Rs. 40,140`
D. ` Rs. 8028`

Explanation

To solve the problem step by step, we will follow the logic provided in the video transcript.

### Step 1: Understand the relationship between cost and weight
The cost of the precious stone varies as the cube of its weight. If we denote the weight of the stone as \( w \), then the cost \( C \) can be expressed as:
\[ C = k \cdot w^3 \]
where \( k \) is a constant of proportionality.

### Step 2: Determine the weights of the broken pieces
The stone breaks into three pieces with weights in the ratio \( 1:2:3 \). Let the weights of the pieces be:
- Weight of piece 1 = \( x \)
- Weight of piece 2 = \( 2x \)
- Weight of piece 3 = \( 3x \)

The total weight of the stone is:
\[ w = x + 2x + 3x = 6x \]

### Step 3: Calculate the original cost of the stone
The original cost \( C_1 \) of the stone can be calculated using the total weight:
\[ C_1 = k \cdot (6x)^3 = k \cdot 216x^3 \]

### Step 4: Set the original cost equal to the given cost
We know from the problem that the cost of the unbroken stone is Rs. 96,336:
\[ 216k \cdot x^3 = 96,336 \]

### Step 5: Solve for \( k \cdot x^3 \)
To find \( k \cdot x^3 \), we divide both sides by 216:
\[ k \cdot x^3 = \frac{96,336}{216} \]
Calculating this gives:
\[ k \cdot x^3 = 446 \]

### Step 6: Calculate the new cost after breakage
Now, we need to find the new cost \( C_2 \) after the stone is broken. The new cost is the sum of the costs of the individual pieces:
\[ C_2 = k \cdot (x^3 + (2x)^3 + (3x)^3) \]
Calculating each term:
- \( (2x)^3 = 8x^3 \)
- \( (3x)^3 = 27x^3 \)

Thus,
\[ C_2 = k \cdot (x^3 + 8x^3 + 27x^3) = k \cdot 36x^3 \]

### Step 7: Substitute \( k \cdot x^3 \) into the new cost
Now substituting \( k \cdot x^3 = 446 \):
\[ C_2 = k \cdot 36x^3 = 36 \cdot 446 = 16,056 \]

### Step 8: Calculate the loss incurred due to breakage
The loss incurred due to breakage can be calculated by subtracting the new cost from the original cost:
\[ \text{Loss} = C_1 - C_2 = 96,336 - 16,056 \]
Calculating this gives:
\[ \text{Loss} = 80,280 \]

### Final Answer
The loss incurred due to breakage is Rs. 80,280.

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