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

Options

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

Explanation

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

### 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) 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 mirror formula to find the image distance (v)
The mirror formula is:
\[ \frac{1}{F} = \frac{1}{v} + \frac{1}{u} \]
Rearranging the formula to find v:
\[ \frac{1}{v} = \frac{1}{F} - \frac{1}{u} \]
Substituting the values of F and u:
\[ \frac{1}{v} = \frac{1}{12.5} - \frac{1}{-40} \]
Calculating the right-hand side:
\[ \frac{1}{v} = \frac{1}{12.5} + \frac{1}{40} \]
Finding a common denominator (which is 200):
\[ \frac{1}{v} = \frac{16}{200} + \frac{5}{200} = \frac{21}{200} \]
Now, taking the reciprocal to find v:
\[ v = \frac{200}{21} \, \text{cm} \]

### Step 4: Calculate the magnification (m)
The magnification formula is:
\[ m = -\frac{v}{u} = \frac{h_i}{h_o} \]
Substituting the values of v and u:
\[ m = -\frac{\frac{200}{21}}{-40} = \frac{200}{21 \times 40} = \frac{200}{840} = \frac{5}{21} \]

### Step 5: Calculate the height of the image (h_i)
Using the magnification:
\[ h_i = m \times h_o \]
Substituting the values:
\[ h_i = \frac{5}{21} \times 7.5 \]
Calculating:
\[ h_i = \frac{5 \times 7.5}{21} = \frac{37.5}{21} \approx 1.7857 \, \text{cm} \]

### Step 6: Final answer
The height of the image is approximately:
\[ h_i \approx 1.79 \, \text{cm} \]