Step 1: Understanding the Concept:
This problem requires calculating the position of an image formed by a curved spherical mirror. To find the correct distance and direction, we apply the standard mirror formula alongside Cartesian sign conventions. Under standard sign conventions, distances measured in the direction of the incoming light are positive, while distances measured against it are negative.
Step 2: Key Formula or Approach:
The standard spherical mirror formula relates focal length ($f$), object distance ($u$), and image distance ($v$):
$$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$
Let's apply standard Cartesian sign conventions for a real object in front of a concave mirror:
- The focal length ($f$) of a concave mirror lies in front of its reflective surface: $f = -20 \text{ cm}$
- The real object ($u$) is placed in front of the reflective surface: $u = -30 \text{ cm}$
Step 3: Detailed Explanation:
Let's isolate the target variable ($\frac{1}{v}$) in the mirror equation:
$$ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} $$
Substitute our signed numerical values into the equation:
$$ \frac{1}{v} = \frac{1}{-20} - \frac{1}{-30} $$
$$ \frac{1}{v} = -\frac{1}{20} + \frac{1}{30} $$
Find a common denominator for the fractions on the right side (which is 60):
$$ \frac{1}{v} = \frac{-3 + 2}{60} $$
$$ \frac{1}{v} = \frac{-1}{60} $$
Inverting both sides of the equation gives the final image distance:
$$ v = -60 \text{ cm} $$
The negative sign indicates that the image forms $60 \text{ cm}$ in front of the mirror on the same side as the object (making it a real and inverted image). The absolute magnitude of the distance matches option (A).
Step 4: Final Answer:
The image distance from the mirror is 60 cm.