The cost of the marble varies directly with square of its weight. Marble is broken into 3 parts whose weights are in the ratio 3 : 4 : 5. If marble had been broken into three equal parts by weight then there would have been a further loss of ₹ 1800. What is the actual cost of the original (or unbroken) marble?

Options

A. ₹ 3600
B. ₹ 10800
C. ₹ 2160
D. none of these

Answer

To solve the problem, we need to determine the actual cost of the original (unbroken) marble based on the information provided. Let's break it down step by step.

### Step 1: Understand the relationship between cost and weight
The cost of the marble varies directly with the square of its weight. This means if the weight of the marble is \( w \), then the cost \( C \) can be expressed as:
\[ C = k \cdot w^2 \]
where \( k \) is a constant of proportionality.

### Step 2: Define the weights of the broken parts
The marble is broken into three parts with weights in the ratio 3:4:5. Let's denote the weights of these parts as:
- Weight 1: \( 3x \)
- Weight 2: \( 4x \)
- Weight 3: \( 5x \)

### Step 3: Calculate the total weight
The total weight of the marble is:
\[ 3x + 4x + 5x = 12x \]

### Step 4: Calculate the cost of each part based on their weights
Using the relationship \( C = k \cdot w^2 \), we can calculate the cost of each part:
- Cost of part 1:
\[ C_1 = k \cdot (3x)^2 = k \cdot 9x^2 \]
- Cost of part 2:
\[ C_2 = k \cdot (4x)^2 = k \cdot 16x^2 \]
- Cost of part 3:
\[ C_3 = k \cdot (5x)^2 = k \cdot 25x^2 \]

### Step 5: Calculate the total cost of the broken marble
The total cost of the broken marble is:
\[ C_{total} = C_1 + C_2 + C_3 = k \cdot 9x^2 + k \cdot 16x^2 + k \cdot 25x^2 = k \cdot (9 + 16 + 25)x^2 = k \cdot 50x^2 \]

### Step 6: Consider the scenario of equal parts
If the marble were broken into three equal parts, each part would weigh:
\[ \frac{12x}{3} = 4x \]
The cost of each equal part would be:
\[ C_{equal} = k \cdot (4x)^2 = k \cdot 16x^2 \]
Thus, the total cost for three equal parts would be:
\[ C_{equal\ total} = 3 \cdot C_{equal} = 3 \cdot k \cdot 16x^2 = k \cdot 48x^2 \]

### Step 7: Set up the equation based on the loss
According to the problem, breaking the marble into equal parts would result in a loss of ₹1800. Therefore, we can set up the equation:
\[ C_{total} - C_{equal\ total} = 1800 \]
Substituting the costs we found:
\[ k \cdot 50x^2 - k \cdot 48x^2 = 1800 \]
This simplifies to:
\[ k \cdot 2x^2 = 1800 \]
Thus, we can solve for \( k \cdot x^2 \):
\[ k \cdot x^2 = \frac{1800}{2} = 900 \]

### Step 8: Calculate the actual cost of the original marble
Now we can find the total cost of the original marble:
\[ C_{original} = k \cdot (12x)^2 = k \cdot 144x^2 \]
Substituting \( k \cdot x^2 = 900 \):
\[ C_{original} = k \cdot 144x^2 = 144 \cdot 900 = 129600 \]

### Final Answer
The actual cost of the original (unbroken) marble is ₹129600.

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