Problem: An object is placed in front of a convex mirror of focal length 30 cm. The image formed by the mirror is a quarter of the size of the object. Find the distance of the object from the mirror.

Solution:

Step 1: Identify the given information

- Focal length of convex mirror (f) = 30 cm
- Size of image (h') = 1/4 x Size of object (h)

Step 2: Use mirror formula to find the distance of the object from the mirror

The mirror formula is given by:

1/f = 1/v + 1/u

where,
- f = focal length of the mirror
- u = distance of the object from the mirror
- v = distance of the image from the mirror

Since the image formed by the mirror is a quarter of the size of the object, we can write:

h'/h = -v/u = -1/4

This gives us:

v = -h'/4
and
u = -h/4

Substituting these values in the mirror formula, we get:

1/30 = -1/(h'/4) - 1/(h/4)

Simplifying this equation, we get:

h' + h = 120

Step 3: Solve for the distance of the object from the mirror

Since the object is placed in front of the mirror, the distance of the object from the mirror (u) is positive. We can use the equation h' + h = 120 to solve for the distance of the object from the mirror.

Let's assume the size of the object is h = x cm. Then, the size of the image is h' = x/4 cm.

Substituting these values in the equation h' + h = 120, we get:

x/4 + x = 120

Simplifying this equation, we get:

5x/4 = 120

x = 96 cm

Therefore, the distance of the object from the mirror is:

u = -h/4 = -96/4 = -24 cm

Since the object is placed in front of the mirror, the distance of the object from the mirror (u) is positive. Therefore, the object is placed at a distance of 24 cm from the convex mirror.

Answer: The distance of the object from the convex mirror is 24 cm.