The cost of a piece of diamond varies with the square of its weight. A diamond of Rs. 5,184 value is cut into 3 pieces whose weights are in the ratio 1:2:3. Find the loss involved in the cutting.

Options

A. Rs. 3,068
B. Rs. 3,088
C. Rs. 3,175
D. Rs. 3,168

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Weight Ratio
The diamond is cut into three pieces with weights in the ratio 1:2:3. Let's denote the weights of the pieces as:
- Weight of first piece = \( x \)
- Weight of second piece = \( 2x \)
- Weight of third piece = \( 3x \)

### Step 2: Calculate Total Weight
The total weight of the diamond is the sum of the weights of the three pieces:
\[ \text{Total weight} = x + 2x + 3x = 6x \]

### Step 3: Understand the Cost Relation
The cost of the diamond varies with the square of its weight. Therefore, if we denote the cost of the diamond as \( C \), we can express the cost in terms of the weights:
- Cost of first piece = \( k \cdot (x^2) \)
- Cost of second piece = \( k \cdot (2x)^2 = k \cdot 4x^2 \)
- Cost of third piece = \( k \cdot (3x)^2 = k \cdot 9x^2 \)

Where \( k \) is a constant of proportionality.

### Step 4: Calculate Total Cost of the Original Diamond
The total cost of the diamond before cutting it is:
\[ C = k \cdot (x^2 + 4x^2 + 9x^2) = k \cdot 14x^2 \]

### Step 5: Set Up the Equation
We know from the problem statement that the total cost of the diamond is Rs. 5,184. Thus, we can set up the equation:
\[ k \cdot 14x^2 = 5184 \]

### Step 6: Find the Value of \( k \cdot x^2 \)
To find \( k \cdot x^2 \), we can rearrange the equation:
\[ k \cdot x^2 = \frac{5184}{14} = 370.2857 \quad (\text{approximately}) \]

### Step 7: Calculate the Cost After Cutting
After cutting, the cost of the diamond pieces is:
\[ \text{Cost after cutting} = k \cdot (x^2 + 4x^2 + 9x^2) = k \cdot 14x^2 = 5184 \]

### Step 8: Calculate the Loss
The loss incurred from cutting the diamond is the difference between the original cost and the cost after cutting. Since the cost after cutting remains the same as the original cost, we need to calculate the loss based on the weight ratios:
- The weights of the pieces are in the ratio 1:2:3, so the effective cost after cutting will be:
\[ \text{Cost after cutting} = k \cdot (1^2 + 2^2 + 3^2) = k \cdot (1 + 4 + 9) = k \cdot 14 \]

### Step 9: Calculate the Loss in Terms of Units
The loss in terms of units is:
\[ \text{Loss} = \text{Original cost} - \text{Cost after cutting} = 5184 - \text{Cost of pieces} \]

### Step 10: Calculate the Value of the Loss
To find the value of the loss, we need to calculate the difference between the original cost and the new cost:
\[ \text{Loss} = 5184 - \text{Cost of pieces} \]

Since the cost of pieces is less than the original cost, we can calculate the loss as follows:
\[ \text{Loss} = 5184 - \left( \frac{5184}{36} \times 22 \right) \]

### Step 11: Final Calculation
The value of 1 unit is:
\[ \text{Value of 1 unit} = \frac{5184}{36} = 144 \]
The loss in rupees is:
\[ \text{Loss} = 144 \times 22 = 3168 \]

### Final Answer
The loss involved in the cutting of the diamond is **Rs. 3,168**.

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