An object of size 7.5 cm is placed in front of a convex mirror of radius of curvature 25 cm at a distance of 40 cm. The size of the image should be

Options

A. 2.3 cm
B. 1.78 cm
C. 1 cm
D. 0.8 cm

Explanation

To solve the problem step by step, we will use the mirror formula and the concept of magnification for a convex mirror.

### Step 1: Identify the given values
- Height of the object (h_o) = 7.5 cm
- Radius of curvature (R) = 25 cm
- Object distance (u) = -40 cm (negative because the object is in front of the mirror)

### Step 2: Calculate the focal length (f) of the convex mirror
The focal length (f) of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{25 \, \text{cm}}{2} = 12.5 \, \text{cm} \]

### Step 3: Use the magnification formula
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = \frac{f}{f - u} \]
Where:
- \( h_i \) = height of the image
- \( h_o \) = height of the object

### Step 4: Substitute the values into the magnification formula
We need to find \( h_i \):
\[ h_i = h_o \cdot \frac{f}{f - u} \]
Substituting the known values:
\[ h_i = 7.5 \cdot \frac{12.5}{12.5 - (-40)} \]
\[ h_i = 7.5 \cdot \frac{12.5}{12.5 + 40} \]
\[ h_i = 7.5 \cdot \frac{12.5}{52.5} \]

### Step 5: Calculate \( h_i \)
Now we calculate:
\[ h_i = 7.5 \cdot \frac{12.5}{52.5} \]
\[ h_i = 7.5 \cdot 0.2381 \] (approximately)
\[ h_i \approx 1.7857 \, \text{cm} \]

### Step 6: Determine the sign of the image height
Since it's a convex mirror, the image formed is virtual and upright, thus:
\[ h_i \approx -1.7857 \, \text{cm} \] (the negative sign indicates that the image is virtual)

### Step 7: Round the value
Rounding \( h_i \) to two decimal places gives:
\[ h_i \approx -1.79 \, \text{cm} \]

### Final Answer
The size of the image is approximately **1.79 cm** (the negative sign indicates that it is virtual).

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