An object 2 cm high is placed at a distance of 16 cm from a concave mirror, which produces 3 cm high inverted image. What is the focal length of the mirror? Also, find the position of the image.

Explanation

To solve the problem step-by-step, we will use the mirror formula and the magnification formula.

### Step 1: Identify the given values
- Height of the object (H1) = 2 cm
- Distance of the object from the mirror (U) = -16 cm (negative because the object is in front of the mirror)
- Height of the image (H2) = -3 cm (negative because the image is inverted)

### Step 2: Use the magnification formula
The magnification (m) is given by the formula:
\[ m = \frac{H2}{H1} = \frac{-V}{U} \]
Substituting the known values:
\[ \frac{-3}{2} = \frac{-V}{-16} \]
This simplifies to:
\[ \frac{3}{2} = \frac{V}{16} \]

### Step 3: Solve for V
Cross-multiplying gives:
\[ 3 \times 16 = 2 \times V \]
\[ 48 = 2V \]
\[ V = \frac{48}{2} = 24 \, \text{cm} \]
Since we are dealing with a concave mirror, we take V as negative:
\[ V = -24 \, \text{cm} \]

### Step 4: Use the mirror formula to find the focal length (f)
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of V and U:
\[ \frac{1}{f} = \frac{1}{-24} + \frac{1}{-16} \]

### Step 5: Find a common denominator and simplify
The common denominator of 24 and 16 is 48. Therefore:
\[ \frac{1}{-24} = \frac{-2}{48} \]
\[ \frac{1}{-16} = \frac{-3}{48} \]
So:
\[ \frac{1}{f} = \frac{-2 - 3}{48} = \frac{-5}{48} \]

### Step 6: Solve for f
Taking the reciprocal gives:
\[ f = \frac{-48}{5} = -9.6 \, \text{cm} \]

### Final Answers
- The focal length of the mirror is **-9.6 cm**.
- The position of the image is **-24 cm** (indicating that the image is formed 24 cm in front of the mirror).