If the side of a cube increases by 15%, then what will be the percentage increase in the volume of the cube?

Options

A. 32.25
B. 30
C. 45
D. 52.08

Answer

To find the percentage increase in the volume of a cube when the side length increases by 15%, we can follow these steps:

### Step 1: Understand the relationship between side length and volume
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of a side of the cube.

### Step 2: Calculate the new side length after the increase
If the side length increases by 15%, the new side length \( a' \) can be calculated as:
\[ a' = a + 0.15a = 1.15a \]

### Step 3: Calculate the new volume
Using the new side length, the new volume \( V' \) can be calculated as:
\[ V' = (a')^3 = (1.15a)^3 \]
Expanding this gives:
\[ V' = 1.15^3 \cdot a^3 \]

### Step 4: Calculate \( 1.15^3 \)
Now we need to calculate \( 1.15^3 \):
\[ 1.15^3 = 1.15 \times 1.15 \times 1.15 = 1.520875 \]

### Step 5: Calculate the percentage increase in volume
The percentage increase in volume can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{V' - V}{V} \right) \times 100 \]
Substituting \( V' = 1.520875 \cdot V \):
\[ \text{Percentage Increase} = \left( \frac{1.520875V - V}{V} \right) \times 100 = (1.520875 - 1) \times 100 \]
\[ = 0.520875 \times 100 = 52.0875\% \]

### Step 6: Round to two decimal places
Rounding \( 52.0875\% \) gives approximately \( 52.08\% \).

### Final Answer
Thus, the percentage increase in the volume of the cube is approximately **52.08%**.

---