A point source is kept at a distance of 1000 m has an illumination I. To change the illumination to 16I the new distance should become

Options

A. a.250 m
B. b.500 m
C. c.750 m
D. d.800 m

Explanation

To solve the problem, we need to understand the relationship between illumination (I) and distance (R) from a point source. The illumination (I) from a point source is inversely proportional to the square of the distance (R) from the source. This can be expressed mathematically as:

\[ I \propto \frac{1}{R^2} \]

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Let the initial illumination be \( I_1 = I \).
- The initial distance from the source is \( R_1 = 1000 \, \text{m} \).

2. **Identify Final Conditions**:
- The final illumination is \( I_2 = 16I \).
- We need to find the new distance \( R_2 \).

3. **Set Up the Proportionality Relationship**:
- From the inverse square law of illumination, we can write:
\[ \frac{I_1}{I_2} = \frac{R_2^2}{R_1^2} \]

4. **Substitute Known Values**:
- Substitute \( I_1 = I \) and \( I_2 = 16I \):
\[ \frac{I}{16I} = \frac{R_2^2}{(1000)^2} \]
- This simplifies to:
\[ \frac{1}{16} = \frac{R_2^2}{1000^2} \]

5. **Cross Multiply to Solve for \( R_2^2 \)**:
\[ R_2^2 = \frac{1000^2}{16} \]

6. **Calculate \( R_2^2 \)**:
- Calculate \( 1000^2 = 1000000 \):
\[ R_2^2 = \frac{1000000}{16} = 62500 \]

7. **Find \( R_2 \)**:
- Take the square root of both sides:
\[ R_2 = \sqrt{62500} = 250 \, \text{m} \]

### Final Answer:
The new distance \( R_2 \) should be **250 m**.

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