The difference in volumes of two cubes in `152 m^3` and the difference in their one face areas is `20 m^2`. If the sum of their edges is 10 m, the product of their edges is -----

Options

A. 21
B. 24
C. 36
D. 48

Answer

To solve the problem step by step, we will denote the edges of the two cubes as \( A \) and \( B \).

### Step 1: Set up the equations based on the given information.

1. The difference in volumes of the two cubes is given as:
\[ A^3 - B^3 = 152 \quad \text{(1)} \]

2. The difference in their face areas is given as:
\[ A^2 - B^2 = 20 \quad \text{(2)} \]

3. The sum of their edges is given as:
\[ A + B = 10 \quad \text{(3)} \]

### Step 2: Use the identity for the difference of cubes.

From equation (1), we can use the identity for the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]

### Step 3: Use the identity for the difference of squares.

From equation (2), we can use the identity for the difference of squares:
\[ A^2 - B^2 = (A - B)(A + B) \]

### Step 4: Substitute \( A + B \) from equation (3) into equation (2).

Using equation (3) in equation (2):
\[ A^2 - B^2 = (A - B)(10) = 20 \]
This simplifies to:
\[ A - B = 2 \quad \text{(4)} \]

### Step 5: Solve equations (3) and (4) simultaneously.

Now we have two equations:
1. \( A + B = 10 \) (from equation (3))
2. \( A - B = 2 \) (from equation (4))

Adding these two equations:
\[ (A + B) + (A - B) = 10 + 2 \]
This simplifies to:
\[ 2A = 12 \implies A = 6 \]

Now substitute \( A \) back into equation (3):
\[ 6 + B = 10 \implies B = 4 \]

### Step 6: Calculate the product of the edges.

Now that we have \( A = 6 \) and \( B = 4 \), we can find the product of their edges:
\[ A \cdot B = 6 \cdot 4 = 24 \]

### Final Answer:
The product of their edges is \( 24 \).

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