The concave and convex surface of a thin concavo-convex lens of refractive index 1.5 has radius of curvature 50 cm and 10 cm respectively. The concave side is slivered then if the equivalent focal length of the silvered lends is `-5x`cm then find x.

Explanation

To solve the problem, we need to find the equivalent focal length of a silvered concavo-convex lens with given parameters. Let's break down the solution step by step.

### Step 1: Identify the parameters
- Refractive index (μ) = 1.5
- Radius of curvature of the convex surface (R1) = +10 cm (positive because it is convex)
- Radius of curvature of the concave surface (R2) = -50 cm (negative because it is concave)

### Step 2: Calculate the focal length of the lens
Using the lens maker's formula:
\[ \frac{1}{f} = (μ - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Substituting the values:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{-50} \right) \]
\[ = 0.5 \left( \frac{1}{10} + \frac{1}{50} \right) \]

### Step 3: Simplify the expression
Finding a common denominator for the fractions:
\[ \frac{1}{10} = \frac{5}{50} \]
Thus,
\[ \frac{1}{10} + \frac{1}{50} = \frac{5}{50} + \frac{1}{50} = \frac{6}{50} = \frac{3}{25} \]

Now substituting back:
\[ \frac{1}{f} = 0.5 \cdot \frac{3}{25} = \frac{3}{50} \]

### Step 4: Calculate the focal length (f)
Taking the reciprocal:
\[ f = \frac{50}{3} \text{ cm} \approx 16.67 \text{ cm} \]

### Step 5: Consider the silvered concave surface
When the concave surface is silvered, it behaves as a mirror with a focal length given by:
\[ f_m = -\frac{R}{2} = -\frac{-50}{2} = -25 \text{ cm} \]

### Step 6: Combine the lens and mirror
The equivalent focal length (F) of the system (lens + mirror) can be calculated using:
\[ \frac{1}{F} = \frac{1}{f} + \frac{1}{f_m} \]
Substituting the values:
\[ \frac{1}{F} = \frac{3}{50} + \frac{1}{-25} \]
Finding a common denominator:
\[ \frac{1}{-25} = -\frac{2}{50} \]
Thus,
\[ \frac{1}{F} = \frac{3}{50} - \frac{2}{50} = \frac{1}{50} \]

### Step 7: Calculate the equivalent focal length (F)
Taking the reciprocal:
\[ F = 50 \text{ cm} \]

### Step 8: Relate to the given focal length
According to the problem, the equivalent focal length of the silvered lens is given as:
\[ F = -5x \]
Setting the two expressions equal:
\[ 50 = -5x \]
Solving for x:
\[ x = -\frac{50}{5} = -10 \]

### Final Answer
The value of \( x \) is \(-10\).

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