A square object is placed `15 cm` from a convex mirror of radius of curavture `90 cm`. Calculate the position of the image and its areal magnification.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Object distance (u) = -15 cm (the negative sign indicates that the object is in front of the mirror)
- Radius of curvature (R) = 90 cm

### Step 2: Calculate the focal length (f)
The focal length (f) of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{90 \, \text{cm}}{2} = 45 \, \text{cm} \]
Since it is a convex mirror, the focal length is positive:
\[ f = +45 \, \text{cm} \]

### Step 3: Apply the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]

### Step 4: Substitute the values into the mirror formula
Substituting the values of f and u:
\[ \frac{1}{v} = \frac{1}{45} - \frac{1}{-15} \]
This simplifies to:
\[ \frac{1}{v} = \frac{1}{45} + \frac{1}{15} \]

### Step 5: Find a common denominator and simplify
The common denominator for 45 and 15 is 45:
\[ \frac{1}{15} = \frac{3}{45} \]
So:
\[ \frac{1}{v} = \frac{1}{45} + \frac{3}{45} = \frac{4}{45} \]

### Step 6: Calculate v
Taking the reciprocal gives:
\[ v = \frac{45}{4} \, \text{cm} = 11.25 \, \text{cm} \]
This is the position of the image.

### Step 7: Calculate the magnification (M)
The magnification (M) is given by the formula:
\[ M = -\frac{v}{u} \]
Substituting the values of v and u:
\[ M = -\frac{\frac{45}{4}}{-15} = \frac{45}{4 \times 15} = \frac{45}{60} = \frac{3}{4} \]

### Final Answers:
- Position of the image (v) = 11.25 cm
- Areal magnification (M) = 0.75 or \( \frac{3}{4} \)