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

Answer

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

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

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

### Step 3: Use the mirror formula
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{12.5} = \frac{1}{v} + \frac{1}{-40} \]

### Step 4: Solve for v
Rearranging the formula to find \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{12.5} + \frac{1}{40} \]
Finding a common denominator (which is 200):
\[ \frac{1}{12.5} = \frac{16}{200} \]
\[ \frac{1}{40} = \frac{5}{200} \]
Adding these fractions:
\[ \frac{1}{v} = \frac{16 + 5}{200} = \frac{21}{200} \]
Now, taking the reciprocal to find \( v \):
\[ v = \frac{200}{21} \approx 9.52 \, \text{cm} \]

### Step 5: Calculate the magnification (m)
The magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the values of \( v \) and \( u \):
\[ m = -\frac{9.52}{-40} = \frac{9.52}{40} \approx 0.238 \]

### Step 6: Find the size of the image (h_i)
Using the magnification to find the height of the image:
\[ h_i = m \cdot h_o \]
Substituting the values:
\[ h_i = 0.238 \times 7.5 \approx 1.785 \, \text{cm} \]

### Final Answer
The size of the image is approximately **1.79 cm**.

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