How far should an object be from a concave spherical mirror of radius 36 cm to form a real
image one-ninth its size?

Explanation

To solve the problem step by step, we will follow the concepts of optics related to concave mirrors, magnification, and the mirror formula.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Radius of curvature (R) = 36 cm
- Since it is a concave mirror, we take R as negative: \( R = -36 \) cm.
- The focal length (f) of a mirror is given by the formula \( f = \frac{R}{2} \). Therefore, \( f = \frac{-36}{2} = -18 \) cm.
- The magnification (m) is given as \( \frac{1}{9} \) of the object size, and since it is a real image, the magnification will be negative: \( m = -\frac{1}{9} \).

2. **Relate Magnification to Object and Image Distance:**
- The magnification (m) is also defined as \( m = \frac{-v}{u} \), where \( v \) is the image distance and \( u \) is the object distance.
- From the magnification formula, we can express \( v \) in terms of \( u \):
\[ -\frac{1}{9} = \frac{-v}{u} \implies v = \frac{1}{9} u \]

3. **Use the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
- Substituting the focal length \( f = -18 \) cm and the expression for \( v \):
\[ \frac{1}{-18} = \frac{1}{u} + \frac{1}{\left(\frac{1}{9}u\right)} \]

4. **Simplify the Equation:**
- The term \( \frac{1}{\left(\frac{1}{9}u\right)} \) simplifies to \( \frac{9}{u} \):
\[ \frac{1}{-18} = \frac{1}{u} + \frac{9}{u} = \frac{10}{u} \]
- Therefore, we have:
\[ \frac{1}{-18} = \frac{10}{u} \]

5. **Cross-Multiply to Solve for \( u \):**
- Cross-multiplying gives:
\[ -18 \cdot 10 = u \implies u = -180 \text{ cm} \]

6. **Conclusion:**
- The object should be placed at a distance of \( 180 \) cm in front of the concave mirror (the negative sign indicates the direction towards the mirror).

### Final Answer:
The object should be placed at a distance of **180 cm** from the concave spherical mirror.