If the edge of a cube is increased by 10%, then what will be the percentage increase in its volume ?

Options

A. 30
B. `31.3`
C. `136.1`
D. `33.1`

Answer

To solve the problem of finding the percentage increase in the volume of a cube when its edge is increased by 10%, we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of an edge of the cube.

### Step 2: Determine the Original Volume
Let the original edge length of the cube be \( a \). Therefore, the original volume \( V_1 \) is:
\[ V_1 = a^3 \]

### Step 3: Calculate the New Edge Length
If the edge length is increased by 10%, the new edge length \( a' \) can be calculated as:
\[ a' = a + 0.1a = 1.1a \]

### Step 4: Calculate the New Volume
Now, we can find the new volume \( V_2 \) using the new edge length:
\[ V_2 = (a')^3 = (1.1a)^3 = 1.1^3 \cdot a^3 \]

### Step 5: Calculate \( 1.1^3 \)
Calculating \( 1.1^3 \):
\[ 1.1^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]
Thus, the new volume becomes:
\[ V_2 = 1.331a^3 \]

### Step 6: Find the Increase in Volume
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V_2 - V_1 = 1.331a^3 - a^3 = (1.331 - 1)a^3 = 0.331a^3 \]

### Step 7: Calculate the Percentage Increase
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V_1} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{0.331a^3}{a^3} \right) \times 100 = 0.331 \times 100 = 33.1\% \]

### Final Answer
Thus, the percentage increase in the volume of the cube when the edge is increased by 10% is **33.1%**.

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