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 of how far an object should be from a concave spherical mirror of radius 36 cm to form a real image one-ninth its size, we can follow these steps:

### Step 1: Determine the focal length of the mirror
The radius of curvature (R) of the concave mirror is given as 36 cm. The focal length (f) can be calculated using the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{36 \, \text{cm}}{2} = 18 \, \text{cm} \]
Since it is a concave mirror, the focal length will be negative:
\[ f = -18 \, \text{cm} \]

### Step 2: Establish the relationship between object distance (u) and image distance (v)
We know that the image size is one-ninth the size of the object. Therefore, we can express this relationship as:
\[ \frac{v}{u} = -\frac{1}{9} \]
This implies that:
\[ v = -\frac{u}{9} \]

### Step 3: Substitute v in the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of f and v:
\[ \frac{1}{-18} = \frac{1}{-\frac{u}{9}} + \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{-18} = -\frac{9}{u} + \frac{1}{u} \]

### Step 4: Combine the fractions
Combining the fractions on the right side:
\[ \frac{1}{-18} = \frac{-9 + 1}{u} = \frac{-8}{u} \]

### Step 5: Cross-multiply to solve for u
Cross-multiplying gives:
\[ -8 = -\frac{u}{18} \]
Multiplying both sides by -18:
\[ u = 144 \, \text{cm} \]

### Step 6: Find the final object distance
Since we have established that the object distance (u) is negative for a real object in front of a concave mirror, we write:
\[ u = -144 \, \text{cm} \]

### Conclusion
Thus, the object should be placed 144 cm in front of the concave mirror to form a real image one-ninth its size.

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