A precious stone weighing 35 grams worth '12,250 is accidentally dropped and gets broken into two pieces having weights in the ratio of 2 : 5. If the price varies as the square of the weight then find the loss incurred.

Options

A. `'5750`
B. `'6000`
C. `'5500`
D. `'5000`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Understand the relationship between price and weight
The price of the stone varies as the square of its weight. We can express this relationship mathematically as:
\[ \text{Price} = k \cdot W^2 \]
where \( k \) is a constant and \( W \) is the weight of the stone.

### Step 2: Determine the value of \( k \)
We know that the total weight of the stone is 35 grams and its price is 12,250. We can substitute these values into the equation to find \( k \):
\[ 12,250 = k \cdot (35)^2 \]
Calculating \( 35^2 \):
\[ 35^2 = 1225 \]
Now substituting back:
\[ 12,250 = k \cdot 1225 \]
To find \( k \), we divide both sides by 1225:
\[ k = \frac{12,250}{1225} = 10 \]

### Step 3: Determine the weights of the two pieces
The stone breaks into two pieces in the ratio of 2:5. Let the weights of the two pieces be \( 2x \) and \( 5x \). The total weight is:
\[ 2x + 5x = 35 \implies 7x = 35 \implies x = 5 \]
Thus, the weights of the two pieces are:
- First piece: \( 2x = 2 \cdot 5 = 10 \) grams
- Second piece: \( 5x = 5 \cdot 5 = 25 \) grams

### Step 4: Calculate the prices of the two pieces
Now we can calculate the price of each piece using the formula \( \text{Price} = k \cdot W^2 \).

**Price of the first piece (10 grams):**
\[ \text{Price}_1 = 10 \cdot (10)^2 = 10 \cdot 100 = 1000 \]

**Price of the second piece (25 grams):**
\[ \text{Price}_2 = 10 \cdot (25)^2 = 10 \cdot 625 = 6250 \]

### Step 5: Calculate the total price of the broken pieces
The total price of the two broken pieces is:
\[ \text{Total Price} = \text{Price}_1 + \text{Price}_2 = 1000 + 6250 = 7250 \]

### Step 6: Calculate the loss incurred
The loss incurred when the stone broke is the original price minus the total price of the broken pieces:
\[ \text{Loss} = \text{Original Price} - \text{Total Price} = 12,250 - 7250 = 5000 \]

### Final Answer
The loss incurred is **5000**.

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