A lens with a focal length of `16 cm` produces a sharp image of an object in two positions, which are `60 cm` apart. Find the distance from the object to the screen.

Explanation

To solve the problem step by step, we will use the displacement method for a lens. The focal length (f) of the lens is given as 16 cm, and the distance between the two positions (x) where the sharp image is formed is 60 cm. We need to find the distance from the object to the screen (d).

### Step 1: Write down the formula for the displacement method
The formula for the displacement method is given by:
\[ f = \frac{d^2 - x^2}{4d} \]
where:
- \( f \) is the focal length of the lens,
- \( d \) is the distance from the object to the screen,
- \( x \) is the distance between the two positions of the image.

### Step 2: Substitute the known values into the formula
Given:
- \( f = 16 \) cm
- \( x = 60 \) cm

Substituting these values into the formula:
\[ 16 = \frac{d^2 - 60^2}{4d} \]

### Step 3: Simplify the equation
First, calculate \( 60^2 \):
\[ 60^2 = 3600 \]
Now the equation becomes:
\[ 16 = \frac{d^2 - 3600}{4d} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 16 \cdot 4d = d^2 - 3600 \]
This simplifies to:
\[ 64d = d^2 - 3600 \]

### Step 5: Rearrange the equation into standard quadratic form
Rearranging the equation gives:
\[ d^2 - 64d - 3600 = 0 \]

### Step 6: Use the quadratic formula to solve for d
The quadratic formula is:
\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case:
- \( a = 1 \)
- \( b = -64 \)
- \( c = -3600 \)

Substituting these values into the quadratic formula:
\[ d = \frac{-(-64) \pm \sqrt{(-64)^2 - 4 \cdot 1 \cdot (-3600)}}{2 \cdot 1} \]
Calculating \( b^2 - 4ac \):
\[ (-64)^2 = 4096 \]
\[ 4 \cdot 1 \cdot (-3600) = -14400 \]
So:
\[ 4096 + 14400 = 18496 \]
Now substituting back:
\[ d = \frac{64 \pm \sqrt{18496}}{2} \]

### Step 7: Calculate the square root and solve for d
Calculating \( \sqrt{18496} \):
\[ \sqrt{18496} \approx 136 \]
Now substituting this value back into the equation:
\[ d = \frac{64 \pm 136}{2} \]

### Step 8: Find the two possible values for d
Calculating the two possible values:
1. \( d = \frac{64 + 136}{2} = \frac{200}{2} = 100 \) cm
2. \( d = \frac{64 - 136}{2} = \frac{-72}{2} = -36 \) cm (not physically meaningful)

Thus, the only valid solution is:
\[ d = 100 \text{ cm} \]

### Final Answer
The distance from the object to the screen is **100 cm**.