The intensity I of light​ (measured in​ foot-candles) varies inversely with the square of the distance from the source. Suppose that the intensity of a light from the source at a distance of 4 meters is 0.085 ​foot-candle. Determine the intensity of a light from the source at a distance of 10 meters.

The problem states that the intensity \( I \) of light varies inversely with the square of the distance \( d \) from the light source. Mathematically, this relationship is expressed as:

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

This means we can write the relationship as:

\[
I = \frac{k}{d^2}
\]

where \( k \) is a constant of proportionality.

### Step 1: Find the constant \( k \)
We know the intensity \( I = 0.085 \) foot-candles when the distance \( d = 4 \) meters. Using the formula, we can find \( k \):

\[
0.085 = \frac{k}{4^2}
\]
\[
0.085 = \frac{k}{16}
\]
\[
k = 0.085 \times 16
\]
\[
k = 1.36
\]

### Step 2: Find the intensity at a distance of 10 meters
Now that we know \( k = 1.36 \), we can use the formula to find the intensity when the distance \( d = 10 \) meters:

\[
I = \frac{1.36}{10^2}
\]
\[
I = \frac{1.36}{100}
\]
\[
I = 0.0136 \text{ foot-candles}
\]

Thus, the intensity of the light at a distance of 10 meters is \( 0.0136 \) foot-candles.

Would you like more details on how inverse square law works or have any questions?

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Here are 5 related questions for practice:
1. What would the intensity be at a distance of 2 meters from the light source?
2. If the intensity at 6 meters is 0.03 foot-candles, what is the constant \( k \)?
3. How does the intensity change if the distance is doubled?
4. If the distance is halved, how does the intensity of light change?
5. What is the relationship between light intensity and distance for a point source of light?

**Tip:** For inverse square laws, remember that small changes in distance can lead to large changes in intensity.

Learn how to calculate the intensity of light using the inverse square law. In this problem, the light intensity at a distance of 4 meters is 0.085 foot-candles. Determine the intensity at 10 meters with a step-by-step solution. Ideal for students in Grades 9-12 studying inverse proportions.

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