If the string length of n is wound on the exterior four walls of a cube of side a cm starting at point C and ending at point D exactly above C , making equally spaced 4 turns . The side of the cube is :

Options

A. a. `a=(2n )/(sqrt(255))`
B. b. `a =((n)^(2))/(16) `
C. c. `a=( n)/(sqrt(257))`
D. d. `a=2 sqrt(15) n`

Answer

To solve the problem, we need to find the side length \( a \) of a cube given that a string of length \( n \) is wound around the exterior of the cube in 4 equally spaced turns, starting at point \( C \) and ending at point \( D \), which is directly above \( C \).

### Step-by-Step Solution:

1. **Understand the Problem**: We have a cube with side length \( a \) cm. The string is wound around the four walls of the cube, making 4 turns.

2. **Calculate the Length of the String for One Turn**:
- The string wraps around the four walls of the cube. In one complete turn around the cube, the string covers the height of the cube (which is \( a \)) and the perimeter of the base (which is \( 4a \)).
- Therefore, the length of the string for one complete turn is \( 4a + a = 5a \).

3. **Calculate the Total Length of the String for 4 Turns**:
- Since the string makes 4 turns, the total length of the string is:
\[ \text{Total Length} = 4 \times (5a) = 20a \]

4. **Set Up the Equation**:
- We know that the total length of the string \( n \) is equal to \( 20a \):
\[ n = 20a \]

5. **Rearranging the Equation**:
- To find \( a \), we rearrange the equation:
\[ a = \frac{n}{20} \]

6. **Final Expression**:
- Thus, the side of the cube \( a \) can be expressed in terms of \( n \):
\[ a = \frac{n}{20} \]