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 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 (h_o) = 2.5 cm
- Distance of the object from the mirror (u) = -10 cm (negative as per the sign convention for concave mirrors)
- Radius of curvature (R) = 30 cm

### 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:
\[ f = \frac{30 \, \text{cm}}{2} = 15 \, \text{cm} \]
Since it’s a concave mirror, we take it as negative:
\[ f = -15 \, \text{cm} \]

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

### Step 4: Rearranging the equation to find v
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{-15} - \frac{1}{-10} \]
Finding a common denominator (which is 30):
\[ \frac{1}{v} = -\frac{2}{30} + \frac{3}{30} \]
\[ \frac{1}{v} = \frac{1}{30} \]
Now, taking the reciprocal:
\[ v = 30 \, \text{cm} \]

### Step 5: Calculate the magnification (m)
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Where:
- \( h_i \) = height of the image
- \( h_o \) = height of the object

Substituting the values:
\[ m = -\frac{30}{-10} = 3 \]

### Step 6: Calculate the height of the image (h_i)
Now, substituting the magnification back to find the height of the image:
\[ m = \frac{h_i}{h_o} \]
Thus,
\[ 3 = \frac{h_i}{2.5} \]
Solving for \( h_i \):
\[ h_i = 3 \times 2.5 = 7.5 \, \text{cm} \]

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

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