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This problem is asking for the fractional amount of Strontium-90 remaining as a function of time, based on an exponential decay model. The substance decays at a rate of 2% per year, and at time \( t = 0 \), 100% of the substance remains.

The general formula for exponential decay is:

\[
P(t) = P_0 \cdot e^{-kt}
\]

Where:
- \( P(t) \) is the fraction of Strontium-90 remaining after time \( t \) years.
- \( P_0 \) is the initial amount (in this case, 100%, so \( P_0 = 1 \)).
- \( k \) is the decay constant.
- \( t \) is the time in years.

Since the decay rate is 2% per year, \( k \) can be calculated from the fact that after 1 year, \( 98\% \) (or \( 0.98 \)) of the substance remains. Thus:

\[
P(1) = P_0 \cdot e^{-k \cdot 1} = 0.98
\]

Solving for \( k \):

\[
e^{-k} = 0.98
\]
\[
-k = \ln(0.98)
\]
\[
k = -\ln(0.98)
\]

Now that we have \( k \), the function for \( P(t) \) becomes:

\[
P(t) = e^{-k t} = e^{-\ln(0.98) t}
\]

Thus, the final expression for the fraction of Strontium-90 remaining after \( t \) years is:

\[
P(t) = e^{-\ln(0.98) t}
\]

Let me know if you'd like more details or if you have any questions!

Here are 5 related questions:
1. How would the equation change if the decay rate was 5% per year?
2. Can you calculate the half-life of Strontium-90 using this model?
3. How much of the substance would remain after 10 years?
4. What is the significance of the decay constant \( k \) in this equation?
5. How would the decay function change if Strontium-90 had a constant removal rate instead of an exponential decay?

**Tip:** Exponential decay models are commonly used in radioactive decay, population decrease, and financial depreciation, where things decline proportionally to their current value.

Suppose strontium-90 decays at a rate of 2 percent per year. Write the fraction P of strontium remaining as a function of t, measured in years. Assume that at time t = 0 there is 100% remaining.

Learn how to calculate the fraction of Strontium-90 remaining as a function of time using an exponential decay model. This problem demonstrates the decay rate of 2% per year and involves key mathematical concepts like exponential functions and logarithms. Suitable for students in Grades 11-12.

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