An object 2.5 cm high is placed at a distance of 10 cm 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

Answer

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) = 2.5 cm
- Object distance (u) = -10 cm (the negative sign indicates that the object is in front of the mirror)
- Radius of curvature (R) = 30 cm

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

### Step 3: Use the mirror formula to find the image distance (v)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-10} \]

### Step 4: Rearrange the equation to solve for v
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-15} + \frac{1}{10} \]
Finding a common denominator (which is 30):
\[ \frac{1}{v} = -\frac{2}{30} + \frac{3}{30} = \frac{1}{30} \]

### Step 5: Calculate the image distance (v)
Taking the reciprocal:
\[ v = 30 \text{ cm} \]

### Step 6: Calculate the magnification (m)
The magnification (m) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{30}{-10} = 3 \]

### Step 7: Calculate the height of the image (h')
The height of the image (h') can be calculated using the magnification:
\[ m = \frac{h'}{h} \]
Rearranging gives:
\[ h' = m \cdot h \]
Substituting the values:
\[ h' = 3 \cdot 2.5 = 7.5 \text{ cm} \]

### Final Answer
The size of the image is **7.5 cm**.

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