One cubic metre piece of copper is melted and recast into a square cross- section bar , 36 m long. An exact cube is cut off from this bar. If cubic metre of copper cost Rs. 108 , then the cost of this cube is :

Options

A. a. 50 paise
B. b. 75 paise
C. c. one rupee
D. d. 1.50 rupee

Answer

To solve the problem step by step, we will follow the information given in the question and the video transcript.

### Step 1: Understand the volume of copper
We know that we have 1 cubic meter of copper. This is the total volume that will be used to create the square cross-section bar.

**Hint:** Remember that the volume of a cube is calculated as side³.

### Step 2: Set up the dimensions of the bar
The bar has a square cross-section and a length of 36 meters. Let the side length of the square cross-section be 'a' meters. The volume of the bar can be expressed as:
\[ \text{Volume of the bar} = \text{Cross-sectional area} \times \text{Length} = a^2 \times 36 \]

**Hint:** The cross-sectional area of a square is side × side.

### Step 3: Equate the volumes
Since the volume of the bar must equal the volume of the copper we started with, we set up the equation:
\[ a^2 \times 36 = 1 \]
This means that the volume of the bar is equal to the volume of the copper.

**Hint:** You can isolate 'a²' by dividing both sides of the equation by 36.

### Step 4: Solve for 'a'
Now, we solve for 'a²':
\[ a^2 = \frac{1}{36} \]
Taking the square root of both sides gives:
\[ a = \frac{1}{6} \text{ meters} \]

**Hint:** Remember to take the square root to find the side length from the area.

### Step 5: Calculate the volume of the cube cut from the bar
The cube that is cut off from the bar has a volume equal to the side length cubed:
\[ \text{Volume of the cube} = a^3 = \left(\frac{1}{6}\right)^3 = \frac{1}{216} \text{ cubic meters} \]

**Hint:** The volume of a cube is calculated as side³.

### Step 6: Calculate the cost of the cube
The cost of 1 cubic meter of copper is Rs. 108. Therefore, the cost of the cube can be calculated as:
\[ \text{Cost of the cube} = \text{Volume of the cube} \times \text{Cost per cubic meter} \]
\[ \text{Cost of the cube} = \frac{1}{216} \times 108 \]

**Hint:** Multiply the volume of the cube by the cost of 1 cubic meter to find the total cost.

### Step 7: Simplify the cost calculation
Calculating the cost:
\[ \text{Cost of the cube} = \frac{108}{216} = \frac{1}{2} \text{ Rs} \]

**Hint:** Simplifying fractions can help you find the answer more easily.

### Final Answer
The cost of the cube is Rs. 0.5 or 50 paisa.