An ellipse is described by using an endless string which is passed over two pins. If the axes are `6 cm` and `4 cm`, the length of the string and distance between the pins are .........

Explanation

To solve the problem, we need to find two things: the length of the string used to describe the ellipse and the distance between the two pins (the foci of the ellipse).

### Step-by-Step Solution:

1. **Identify the semi-major and semi-minor axes:**
- The lengths of the axes are given as 6 cm and 4 cm.
- The semi-major axis \( a \) is half of the major axis:
\[ a = \frac{6}{2} = 3 \text{ cm} \]
- The semi-minor axis \( b \) is half of the minor axis:
\[ b = \frac{4}{2} = 2 \text{ cm} \]

2. **Write the equation of the ellipse:**
- The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- Substituting the values of \( a \) and \( b \):
\[ \frac{x^2}{3^2} + \frac{y^2}{2^2} = 1 \quad \Rightarrow \quad \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

3. **Calculate the eccentricity \( e \):**
- The eccentricity \( e \) is calculated using the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
- Substituting the values of \( a \) and \( b \):
\[ e = \sqrt{1 - \frac{2^2}{3^2}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

4. **Find the distance between the foci \( F_1 \) and \( F_2 \):**
- The distance between the foci is given by:
\[ 2c = 2ae \]
- Where \( c = ae \). Substituting the values:
\[ 2c = 2 \times 3 \times \frac{\sqrt{5}}{3} = 2\sqrt{5} \text{ cm} \]

5. **Determine the length of the string:**
- The length of the string used to describe the ellipse is equal to the sum of the distances from any point on the ellipse to the two foci, which is constant and equal to \( 2a \):
\[ \text{Length of the string} = 2a = 2 \times 3 = 6 \text{ cm} \]

### Final Answers:
- **Length of the string:** 6 cm
- **Distance between the pins (foci):** \( 2\sqrt{5} \) cm