A convexo-concave convergent lens is made of glass of refractive index 1.5 and focal length `24 cm.` Radius of curvature for one surface is double than that of the other. Then,radii of curvature for the two surfaces are (in cm)

Options

A. `6,12`
B. `12,24`
C. `3,6`
D. `18,36`

Explanation

To solve the problem step by step, we will use the lens maker's formula and the given information about the lens.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have a convexo-concave lens made of glass with a refractive index (μ) of 1.5 and a focal length (f) of +24 cm. The radius of curvature for one surface (R2) is double that of the other surface (R1).

2. **Assign Variables:**
Let:
- R1 = r (the radius of curvature of the first surface)
- R2 = 2r (the radius of curvature of the second surface)

3. **Lens Maker's Formula:**
The lens maker's formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R1} - \frac{1}{R2} \right) \]
Substituting the known values:
- f = +24 cm
- μ = 1.5
- R1 = r
- R2 = 2r

4. **Substituting into the Formula:**
Substitute the values into the lens maker's formula:
\[ \frac{1}{24} = (1.5 - 1) \left( \frac{1}{r} - \frac{1}{2r} \right) \]
Simplifying:
\[ \frac{1}{24} = 0.5 \left( \frac{1}{r} - \frac{1}{2r} \right) \]

5. **Simplifying the Right Side:**
The expression inside the parentheses can be simplified:
\[ \frac{1}{r} - \frac{1}{2r} = \frac{2}{2r} - \frac{1}{2r} = \frac{1}{2r} \]
Therefore, we have:
\[ \frac{1}{24} = 0.5 \cdot \frac{1}{2r} \]
This simplifies to:
\[ \frac{1}{24} = \frac{0.5}{2r} = \frac{1}{4r} \]

6. **Cross-Multiplying:**
Cross-multiplying gives:
\[ 1 \cdot 4r = 24 \cdot 1 \]
Thus:
\[ 4r = 24 \]

7. **Solving for r:**
Dividing both sides by 4:
\[ r = \frac{24}{4} = 6 \text{ cm} \]

8. **Finding R1 and R2:**
Now we can find R1 and R2:
- R1 = r = 6 cm
- R2 = 2r = 2 \times 6 = 12 cm

### Final Answer:
The radii of curvature for the two surfaces are:
- R1 = 6 cm
- R2 = 12 cm