A 3 cm tall object is placed at a distance of 7.5 cm from a convex mirror of focal length 6 cm. Find the location, size and nature of the image.

Explanation

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) = 3 cm (above the principal axis, so it's positive)
- Object distance (u) = -7.5 cm (since it's on the left side of the mirror, it's negative)
- Focal length (f) = +6 cm (for a convex mirror, the focal length is positive)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{6} = \frac{1}{v} + \frac{1}{-7.5} \]
This can be rearranged to find \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{6} + \frac{1}{7.5} \]

### Step 3: Find a common denominator and calculate \( \frac{1}{v} \)
The least common multiple of 6 and 7.5 is 30. We can rewrite the fractions:
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{7.5} = \frac{4}{30} \]
Adding these gives:
\[ \frac{1}{v} = \frac{5}{30} + \frac{4}{30} = \frac{9}{30} \]
Thus,
\[ v = \frac{30}{9} = \frac{10}{3} \text{ cm} \]

### Step 4: Determine the location of the image
Since \( v \) is positive, the image is formed behind the mirror.

### Step 5: Calculate the height of the image using magnification
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the known values:
\[ m = -\frac{\frac{10}{3}}{-7.5} = \frac{10}{3} \times \frac{1}{7.5} = \frac{10}{3} \times \frac{2}{15} = \frac{20}{45} = \frac{4}{9} \]
Now, using the magnification to find the height of the image:
\[ h_i = m \cdot h_o = \frac{4}{9} \cdot 3 = \frac{12}{9} = \frac{4}{3} \text{ cm} \]

### Step 6: Determine the nature of the image
Since the height of the image \( h_i \) is positive, the image is erect.

### Final Answers
- **Location of the image:** \( \frac{10}{3} \) cm behind the mirror
- **Height of the image:** \( \frac{4}{3} \) cm
- **Nature of the image:** Erect