An object `2.5cm` high is placed at a distance of `10cm` from a concave mirror of radius of curvature `30 cm. ` The size of the image is

Options

A. `9.2 cm`
B. `10.5 cm`
C. `5.6 cm`
D. `7.5 cm`

Explanation

To find the size of the image formed by a concave mirror, we can follow these steps:

### Step 1: Identify the Given Values
- Height of the object (h_o) = 2.5 cm
- Object distance (u) = -10 cm (negative because the object is in front of the mirror)
- Radius of curvature (R) = -30 cm (negative for concave mirrors)

### Step 2: Calculate the Focal Length (f)
The focal length (f) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{-30 \, \text{cm}}{2} = -15 \, \text{cm} \]

### Step 3: Use the Mirror Formula
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of f and u:
\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-10} \]

### Step 4: Rearranging the Equation
Rearranging the equation to find \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{-15} + \frac{1}{10} \]

### Step 5: Finding a Common Denominator
The common denominator for -15 and 10 is 30:
\[ \frac{1}{v} = \frac{-2}{30} + \frac{3}{30} = \frac{1}{30} \]

### Step 6: Calculate the Image Distance (v)
Taking the reciprocal gives:
\[ v = 30 \, \text{cm} \]

### Step 7: Calculate the Magnification (m)
The magnification (m) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values of v and u:
\[ m = -\frac{30}{-10} = 3 \]

### Step 8: Calculate the Height of the Image (h_i)
The height of the image can be calculated using the magnification:
\[ h_i = m \cdot h_o \]
Substituting the values:
\[ h_i = 3 \cdot 2.5 \, \text{cm} = 7.5 \, \text{cm} \]

### Conclusion
The size of the image is **7.5 cm**.

---