If the side of a cube be increased by 0.1% find the corresponding increase in the volume of the cube.

Answer

To solve the problem of finding the corresponding increase in the volume of a cube when the side is increased by 0.1%, we can follow these steps:

### Step 1: Define the original side length
Let the original side length of the cube be \( s \).

### Step 2: Calculate the increase in side length
The side length is increased by 0.1%. Therefore, the increase in side length can be calculated as:
\[ \text{Increase} = \frac{0.1}{100} \times s = 0.001s \]

### Step 3: Determine the new side length
The new side length \( s' \) after the increase will be:
\[ s' = s + \text{Increase} = s + 0.001s = 1.001s \]

### Step 4: Calculate the original volume
The volume \( V \) of the cube with side length \( s \) is given by:
\[ V = s^3 \]

### Step 5: Calculate the new volume
The new volume \( V' \) of the cube with the new side length \( s' \) is:
\[ V' = (s')^3 = (1.001s)^3 \]
Using the binomial expansion, we can approximate:
\[ (1.001s)^3 = 1.001^3 \cdot s^3 \]
Calculating \( 1.001^3 \):
\[ 1.001^3 \approx 1 + 3 \times 0.001 + 3 \times (0.001)^2 + (0.001)^3 \approx 1 + 0.003 + 0 + 0 \approx 1.003 \]
Thus,
\[ V' \approx 1.003s^3 \]

### Step 6: Calculate the change in volume
The change in volume \( \Delta V \) is given by:
\[ \Delta V = V' - V = 1.003s^3 - s^3 = (1.003 - 1)s^3 = 0.003s^3 \]

### Step 7: Calculate the percentage increase in volume
The percentage increase in volume can be calculated as:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V} \right) \times 100 = \left( \frac{0.003s^3}{s^3} \right) \times 100 = 0.003 \times 100 = 0.3\% \]

### Final Answer
The corresponding increase in the volume of the cube when the side is increased by 0.1% is **0.3%**.

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