On breaking a cubic solid (edge = 1 m) into fine cubic particles of edge `1mum`, surface becomes `nxx100000`. The value of n is ?

Answer

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

### Step 1: Calculate the volume of the original cubic solid
The edge length of the original cubic solid is given as 1 meter. The volume \( V \) of a cube is calculated using the formula:
\[ V = a^3 \]
where \( a \) is the edge length. Thus,
\[ V = (1 \, \text{m})^3 = 1 \, \text{m}^3 \]

### Step 2: Determine the volume of the fine cubic particles
After breaking the original cube into fine cubic particles with an edge length of \( 1 \, \mu m = 1 \times 10^{-6} \, \text{m} \), the volume of one fine cubic particle is:
\[ V_1 = (1 \times 10^{-6} \, \text{m})^3 = 1 \times 10^{-18} \, \text{m}^3 \]
Let \( n \) be the number of fine cubic particles. The total volume of all fine cubic particles is:
\[ V_{\text{total}} = n \times V_1 = n \times 1 \times 10^{-18} \, \text{m}^3 \]
Since the total volume must remain the same as the original cube, we set:
\[ 1 \, \text{m}^3 = n \times 1 \times 10^{-18} \, \text{m}^3 \]
From this, we can solve for \( n \):
\[ n = \frac{1}{1 \times 10^{-18}} = 10^{18} \]

### Step 3: Calculate the surface area of the original cubic solid
The surface area \( S \) of a cube is given by:
\[ S = 6a^2 \]
For the original cube with edge length 1 meter:
\[ S = 6 \times (1 \, \text{m})^2 = 6 \, \text{m}^2 \]

### Step 4: Calculate the surface area of the fine cubic particles
The surface area \( S_1 \) of one fine cubic particle is:
\[ S_1 = 6a_1^2 \]
where \( a_1 = 1 \times 10^{-6} \, \text{m} \):
\[ S_1 = 6 \times (1 \times 10^{-6} \, \text{m})^2 = 6 \times 10^{-12} \, \text{m}^2 \]
The total surface area of all fine cubic particles is:
\[ S_{\text{total}} = n \times S_1 = 10^{18} \times 6 \times 10^{-12} \, \text{m}^2 = 6 \times 10^{6} \, \text{m}^2 \]

### Step 5: Calculate the factor increase in surface area
The factor increase \( F \) in surface area is given by:
\[ F = \frac{S_{\text{total}}}{S} = \frac{6 \times 10^{6} \, \text{m}^2}{6 \, \text{m}^2} = 10^{6} \]

### Step 6: Relate the factor increase to the given expression
According to the problem, the surface area becomes \( n \times 10^{5} \). We have found that the factor increase is \( 10^{6} \):
\[ n \times 10^{5} = 10^{6} \]
To find \( n \), we divide both sides by \( 10^{5} \):
\[ n = \frac{10^{6}}{10^{5}} = 10 \]

### Final Answer
The value of \( n \) is \( 10 \).

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