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The problem asks about the intensity of light decreasing as it passes through a transparent medium, modeled by an exponential decay equation. Let's break down the problem.

### 1. Given Information:
- The intensity \( I(t) \) decreases proportionally to the thickness \( t \) of the medium in feet.
- The differential equation is: 
  \[
  \frac{dI}{dt} = kI
  \]
  where \( k \) is a constant of proportionality.
- In clear seawater, the intensity \( I \) is 25% of the initial intensity \( I_0 \) at 3 feet below the surface.

### 2. Solving for the Constant \( k \):
The equation given is a separable differential equation. We know:
\[
I(t) = I_0 e^{kt}
\]
At \( t = 3 \) feet, \( I = 0.25 I_0 \) (25% of initial intensity):
\[
0.25 I_0 = I_0 e^{3k}
\]
Dividing both sides by \( I_0 \):
\[
0.25 = e^{3k}
\]
Taking the natural logarithm on both sides:
\[
\ln(0.25) = 3k
\]
\[
k = \frac{\ln(0.25)}{3}
\]
Now, we compute the value of \( k \).

### 3. Finding the Intensity at \( t = 10 \) Feet:
Once we find \( k \), we will use it to find the intensity 10 feet below the surface using the same equation:
\[
I(10) = I_0 e^{10k}
\]
Now, let's compute these values.### 1. The constant of proportionality \( k \) is approximately:
\[
k = -0.46210
\]

### 2. The intensity of the beam 10 feet below the surface is approximately:
\[
I(10) = 0.00984 I_0
\]
This means that the intensity is about 0.984% of the original intensity at 10 feet below the surface.

Do you have any questions or need further clarification?

### Related Questions:
1. How would the intensity change if the depth was doubled from 10 feet to 20 feet?
2. Can you derive a general expression for the intensity at any depth \( t \)?
3. How does the value of \( k \) influence the rate of intensity decay in the medium?
4. What assumptions are made about the medium for this model to hold true?
5. How would the intensity change if the medium were denser, reducing the intensity by 10% per foot?

### Tip:
Always check the units used in the exponential model to ensure consistent results across different applications (feet, meters, etc.).

When a vertical beam of light passes through a transparent medium, the rate at which its intensity decreases is proportional to the intensity. The intensity 3 feet below the surface is 25% of the initial intensity. Find the constant of proportionality and the intensity 10 feet below the surface.

Learn how to solve an exponential decay problem involving light intensity passing through a transparent medium. This problem explores the constant of proportionality and calculates light intensity at 10 feet below the surface. Suitable for advanced high school students (Grades 11-12).

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