An object `5cm` tall is placed `10cm` form a convex mirror of radius of curvature `30cm`. What is the nature and size of the image ?

Options

A. real, `3 cm`
B. virtual, `7.5 cm`
C. virtual, `3 cm`
D. real, `7.5 cm`

Explanation

To solve the problem of determining the nature and size of the image formed by a convex mirror, we will follow these steps:

### Step 1: Identify the given parameters
- Height of the object (h_o) = 5 cm
- Distance of the object from the mirror (u) = -10 cm (negative because the object is in front of the mirror)
- Radius of curvature (R) = 30 cm

### 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:
\[ f = \frac{30 \, \text{cm}}{2} = 15 \, \text{cm} \]
Since it is a convex mirror, the focal length is positive:
\[ f = +15 \, \text{cm} \]

### Step 3: Use the mirror formula to find the image distance (v)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging this formula to find v:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{v} = \frac{1}{15} - \frac{1}{-10} \]
\[ \frac{1}{v} = \frac{1}{15} + \frac{1}{10} \]
To add these fractions, find a common denominator (30):
\[ \frac{1}{v} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} \]
Thus,
\[ v = \frac{30}{5} = 6 \, \text{cm} \]
The positive value indicates that the image is virtual and located behind the mirror.

### Step 4: Calculate the magnification (m)
The magnification (m) is given by the formula:
\[ m = -\frac{v}{u} \]
Substituting the values of v and u:
\[ m = -\frac{6}{-10} = \frac{6}{10} = \frac{3}{5} \]
Since the magnification is positive, this confirms that the image is virtual and upright.

### Step 5: Calculate the size of the image (h_i)
The size of the image can be calculated using the magnification formula:
\[ m = \frac{h_i}{h_o} \]
Rearranging to find h_i:
\[ h_i = m \cdot h_o \]
Substituting the values:
\[ h_i = \frac{3}{5} \cdot 5 \, \text{cm} = 3 \, \text{cm} \]

### Conclusion
The nature of the image is virtual, and the size of the image is 3 cm.

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