The image formed by a convex mirror of focal length 30 cm is a quarter of the object. What is the distance of the object from the mirror?

Explanation

To solve the problem step by step, we will use the concepts of magnification and the mirror formula for a convex mirror.

### Step 1: Understand the given information
- Focal length (f) of the convex mirror = +30 cm (positive because it's a convex mirror)
- The image formed is a quarter of the object, which means the height of the image (h') is 1/4 of the height of the object (h).

### Step 2: Write the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Where:
- \( h' \) = height of the image
- \( h \) = height of the object
- \( v \) = image distance
- \( u \) = object distance

Since the image is a quarter of the object:
\[ m = \frac{1}{4} \]

### Step 3: Relate magnification to object and image distances
From the magnification formula:
\[ \frac{1}{4} = -\frac{v}{u} \]
This implies:
\[ v = -\frac{u}{4} \]

### Step 4: Use the mirror formula
The mirror formula for a convex mirror is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting \( v = -\frac{u}{4} \) into the mirror formula:
\[ \frac{1}{30} = \frac{1}{-\frac{u}{4}} + \frac{1}{u} \]

### Step 5: Simplify the equation
Rearranging the terms:
\[ \frac{1}{30} = -\frac{4}{u} + \frac{1}{u} \]
Combining the fractions on the right side:
\[ \frac{1}{30} = -\frac{4 - 1}{u} \]
\[ \frac{1}{30} = -\frac{3}{u} \]

### Step 6: Solve for u
Cross-multiplying gives:
\[ u = -90 \text{ cm} \]

### Conclusion
The distance of the object from the mirror is 90 cm (the negative sign indicates that the object is on the same side as the incoming light, which is typical for a convex mirror).