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To solve this problem, we can use the concept of inverse square law. The intensity \( I \) of light varies inversely with the square of the distance \( d \), so:

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

This implies:
\[
I = \frac{k}{d^2}
\]
where \( k \) is a constant.

### Step 1: Find the constant \( k \)
Given:
- \( I = 360 \, \text{W/m}^2 \) when \( d = 8 \, \text{m} \).

Substitute these values into the equation to solve for \( k \):

\[
360 = \frac{k}{8^2}
\]
\[
360 = \frac{k}{64}
\]
\[
k = 360 \times 64 = 23040
\]

So, the equation for intensity becomes:
\[
I = \frac{23040}{d^2}
\]

### Step 2: Find the distance \( d \) for \( I = 250 \, \text{W/m}^2 \)
Now, we want to find the distance \( d \) when \( I = 250 \, \text{W/m}^2 \):

\[
250 = \frac{23040}{d^2}
\]
\[
d^2 = \frac{23040}{250}
\]
\[
d^2 = 92.16
\]
\[
d = \sqrt{92.16}
\]
\[
d \approx 9.6 \, \text{m}
\]

### Step 3: Calculate how much farther this point is from the initial distance
The initial distance was \( 8 \, \text{m} \). The new distance is approximately \( 9.6 \, \text{m} \). Thus, the point is:

\[
9.6 - 8 = 1.6 \, \text{m}
\]

### Final Answer:
The point where the intensity is \( 250 \, \text{W/m}^2 \) is approximately \( 1.6 \, \text{m} \) farther from the original point where the intensity was \( 360 \, \text{W/m}^2 \).

Would you like further details or have any questions? 

Here are some related questions you may find interesting:
1. How would the answer change if the initial intensity was higher?
2. What happens to intensity if the distance doubles?
3. Can this inverse square law be applied to other physical quantities?
4. What is the significance of the constant \( k \) in similar problems?
5. How would a different medium, like water, affect light intensity over distance?

**Tip:** Remember, in inverse square law problems, intensity decreases rapidly as distance increases, since it’s proportional to \( \frac{1}{d^2} \).

The intensity I of light from a light bulb varies inversely as the square of the distance d from the light bulb. Suppose I is 360 W/m^2 (watts per square meter) when the distance is 8 m. How much farther would it be to a point where the intensity is 250 W/m^2?

Learn how to calculate the distance where light intensity changes based on the inverse square law. In this problem, intensity varies inversely with the square of the distance from a light source. Given an intensity of 360 W/m^2 at 8 meters, find how much farther it would be for an intensity of 250 W/m^2.

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