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The question provides the function \( f(t) = 500(0.5)^t \), which models the intensity of an X-ray beam as a function of \( t \), the depth in millimeters below the surface of an iron sample. The task is to find the number of particles when \( t = 0 \), which represents the surface of the sample.

To solve:

1. Substitute \( t = 0 \) into the function:
   \[
   f(0) = 500(0.5)^0
   \]

2. Since \( (0.5)^0 = 1 \), this simplifies to:
   \[
   f(0) = 500 \times 1 = 500
   \]

So, the estimated number of particles at the surface is **500**.

The correct answer is **(A) 500**.

Would you like more details, or do you have any questions?

Here are 5 related questions for further practice:
1. What would be the intensity of particles at \( t = 1 \) mm below the surface?
2. How would the intensity change if the base of the exponent was modified?
3. If the initial number of particles were different, say 1000, how would the function change?
4. How does the exponential decay model apply in fields other than physics?
5. How would we interpret a negative value of \( t \) in this function?

**Tip:** Remember that any number raised to the power of zero equals 1, which can simplify calculations at \( t = 0 \) in exponential functions.

The function f models the intensity of an X-ray beam, in number of particles in the X-ray beam, t millimeters below the surface of a sample of iron. According to the model, what is the estimated number of particles in the X-ray beam when it is at the surface of the sample of iron? Given: f(t) = 500(0.5)^t.

Learn how to calculate the intensity of an X-ray beam at the surface of an iron sample using an exponential decay model. This problem involves substitution in an exponential function and covers important concepts in exponential decay. Suitable for high school students in Grades 9-12.

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