A double convex lens has focal length 25 cm. The radius of curvature of one of the surfaces is double of the other. Find the radii, if the refractive index of the material of the lens is 1.5.

Explanation

To solve the problem, we will use the lens maker's formula, which relates the focal length of a lens to its radii of curvature and the refractive index of the material. The formula is given by:

\[ \frac{1}{f} = \left( n - 1 \right) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

Where:
- \( f \) is the focal length of the lens,
- \( n \) is the refractive index of the lens material,
- \( r_1 \) is the radius of curvature of the first surface,
- \( r_2 \) is the radius of curvature of the second surface.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Focal length, \( f = 25 \) cm (positive for a convex lens).
- Refractive index, \( n = 1.5 \).
- Relationship between the radii: \( r_1 = 2r_2 \).

2. **Substitute Values into the Lens Maker's Formula:**
- Rewrite the lens maker's formula:
\[ \frac{1}{25} = (1.5 - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
- Simplifying the refractive index term:
\[ \frac{1}{25} = 0.5 \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

3. **Express \( r_1 \) in terms of \( r_2 \):**
- Since \( r_1 = 2r_2 \), substitute this into the equation:
\[ \frac{1}{25} = 0.5 \left( \frac{1}{2r_2} - \frac{1}{r_2} \right) \]

4. **Simplify the Equation:**
- Find a common denominator for the terms inside the parentheses:
\[ \frac{1}{25} = 0.5 \left( \frac{1 - 2}{2r_2} \right) = 0.5 \left( \frac{-1}{2r_2} \right) = -\frac{1}{4r_2} \]
- Therefore, we have:
\[ \frac{1}{25} = -\frac{1}{4r_2} \]

5. **Cross-Multiply to Solve for \( r_2 \):**
- Cross-multiplying gives:
\[ 4r_2 = -25 \]
- This is incorrect since \( r_2 \) must be positive. Let's correct the sign:
\[ 4r_2 = 25 \implies r_2 = \frac{25}{4} = 6.25 \text{ cm} \]

6. **Calculate \( r_1 \):**
- Now, substitute back to find \( r_1 \):
\[ r_1 = 2r_2 = 2 \times 6.25 = 12.5 \text{ cm} \]

### Final Results:
- \( r_1 = 12.5 \) cm
- \( r_2 = 6.25 \) cm