In a convex mirror having radius of curvature 30 cm the height of image is half the object height. What will be the object (in cm) distance?

The correct answer is 15.

Step-by-step Explanation:

We are given the following parameters for a convex mirror:
1. Radius of curvature, $R=+30\text{cm}$ (positive because the center of curvature of a convex mirror lies behind it).
2. The focal length ($f$) of a mirror is half of its radius of curvature:
$f=\frac{R}{2}=\frac{30}{2}=+15\text{cm}$

Step 1: Using the magnification formula
A convex mirror always forms a virtual, erect, and diminished image behind the mirror. Thus, the magnification ($m$) is positive.
Since the height of the image is half the object height, we have:
$m=+\frac{1}{2}$

The relation for magnification of a spherical mirror is:
$m=-\frac{v}{u}$
where $v$ is the image distance and $u$ is the object distance.

Equating the two expressions for magnification:
$-\frac{v}{u}=\frac{1}{2}$
$v=-\frac{u}{2}$

Step 2: Applying the mirror formula
The mirror formula is given by:
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Substitute the value of $v=-\frac{u}{2}$ and $f=15\text{cm}$ into the formula:
$\frac{1}{-\frac{u}{2}}+\frac{1}{u}=\frac{1}{15}$
$-\frac{2}{u}+\frac{1}{u}=\frac{1}{15}$
$-\frac{1}{u}=\frac{1}{15}$
$u=-15\text{cm}$

The negative sign indicates that the object is placed in front of the mirror (real space). Thus, the distance of the object from the mirror is 15 cm.