An object is 30.0 cm form a spherical mirror, along the central axis.
The absolute value of lateral magnification is `1/2`, The image produced is inverted .
What is the focal length of the mirror?

Explanation

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

### Step 1: Identify the given values
- Object distance (U) = -30 cm (the negative sign indicates that the object is in front of the mirror)
- Lateral magnification (|M|) = 1/2 (since the image is inverted, M will be negative)

### Step 2: Use the magnification formula
The magnification (M) is given by the formula:
\[ M = \frac{V}{U} \]
Where:
- V = image distance
- U = object distance

Since the absolute value of magnification is given as 1/2, we can write:
\[ -\frac{V}{30} = -\frac{1}{2} \]

### Step 3: Solve for V
From the magnification equation, we can rearrange it to find V:
\[ V = \frac{1}{2} \times U \]
Substituting U = -30 cm:
\[ V = \frac{1}{2} \times (-30) = -15 \text{ cm} \]

### Step 4: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Where F is the focal length.

### Step 5: Substitute values into the mirror formula
Substituting V = -15 cm and U = -30 cm into the mirror formula:
\[ \frac{1}{F} = \frac{1}{-15} + \frac{1}{-30} \]

### Step 6: Calculate the right-hand side
Calculating the right-hand side:
\[ \frac{1}{F} = -\frac{1}{15} - \frac{1}{30} \]

To add these fractions, we need a common denominator, which is 30:
\[ \frac{1}{F} = -\frac{2}{30} - \frac{1}{30} = -\frac{3}{30} = -\frac{1}{10} \]

### Step 7: Solve for F
Taking the reciprocal gives:
\[ F = -10 \text{ cm} \]

### Conclusion
The focal length of the mirror is -10 cm, indicating that it is a concave mirror.

---