An object is at a distance of 10 cm from a concave mirror and the image of the object is at a distance of 30 cm from the mirror on the same side as that of the object. The radius of curvature of the concave mirror is

Options

A. `+15.0` cm
B. `+7.5` cm
C. `-7.5` cm
D. `-15.0` cm

Answer

To solve the problem step by step, we will use the mirror formula and the relationship between the focal length and the radius of curvature of a concave mirror.

### Step 1: Understand the given information
- The object distance (U) is given as 10 cm from the mirror. Since the object is in front of the mirror, we take U as negative:
\[ U = -10 \, \text{cm} \]
- The image distance (V) is given as 30 cm from the mirror on the same side as the object. For a concave mirror, real images are also taken as negative:
\[ V = -30 \, \text{cm} \]

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of V and U into the formula:
\[ \frac{1}{f} = \frac{1}{-30} + \frac{1}{-10} \]

### Step 3: Calculate the right-hand side
Finding a common denominator (which is 30):
\[ \frac{1}{f} = -\frac{1}{30} - \frac{3}{30} = -\frac{4}{30} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{2}{15} \]

### Step 4: Find the focal length (f)
Taking the reciprocal to find f:
\[ f = -\frac{15}{2} \, \text{cm} = -7.5 \, \text{cm} \]

### Step 5: Relate the focal length to the radius of curvature (R)
The relationship between the focal length (f) and the radius of curvature (R) for a concave mirror is:
\[ f = \frac{R}{2} \]
Substituting the value of f:
\[ -\frac{15}{2} = \frac{R}{2} \]

### Step 6: Solve for R
Multiplying both sides by 2 to isolate R:
\[ R = -15 \, \text{cm} \]

### Conclusion
The radius of curvature of the concave mirror is:
\[ \boxed{-15 \, \text{cm}} \]