Strontium 90 is a radioactive isotope that decays according to the eqaution At=A0e-0.0244t, where A0 is the initial amount present and At is the amount present after t years. If you begin with 400 grams of strontium 90,

(a) How much strontium 90 will be left after 10 years? Round your answer to the nearest tenth of a gram.

 grams

(b) When will 140 grams of strontium 90 be left? Round your answer to the nearest tenth of a year.

 years

Given the decay equation for strontium-90:

\[
A_t = A_0 \times e^{-0.0244t}
\]

where:
- \( A_0 = 400 \) grams (initial amount),
- \( A_t \) is the amount left after \( t \) years,
- \( k = 0.0244 \) per year (decay constant).

Let's solve the two parts of the problem:

### **(a) Amount left after 10 years**

We need to find \( A_t \) when \( t = 10 \) years.

\[
A_t = 400 \times e^{-0.0244 \times 10}
\]

Now, calculate the exponent and the resulting amount:

\[
A_t = 400 \times e^{-0.244}
\]

Using the value of \( e^{-0.244} \):

\[
A_t \approx 400 \times 0.7838 \approx 313.5 \text{ grams}
\]

So, after 10 years, approximately **313.5 grams** of strontium-90 will be left.

### **(b) Time when 140 grams are left**

We need to find \( t \) when \( A_t = 140 \) grams.

Starting with the equation:

\[
140 = 400 \times e^{-0.0244t}
\]

First, solve for \( e^{-0.0244t} \):

\[
\frac{140}{400} = e^{-0.0244t}
\]

\[
0.35 = e^{-0.0244t}
\]

Take the natural logarithm on both sides:

\[
\ln(0.35) = -0.0244t
\]

\[
t = \frac{\ln(0.35)}{-0.0244}
\]

Now, compute \( t \):

\[
t \approx \frac{-1.0498}{-0.0244} \approx 43.0 \text{ years}
\]

So, it will take approximately **43.0 years** for the amount of strontium-90 to decay to 140 grams.

---

**Follow-Up Questions:**

1. How would the time change if the decay constant were different?
2. Can you explain why the natural logarithm is used in part (b)?
3. How does the decay constant \( k \) affect the speed of radioactive decay?
4. What would happen to the remaining amount after 20 years instead of 10?
5. How does the initial amount \( A_0 \) affect the time it takes to decay to a certain amount?

**Tip:** When solving exponential decay problems, always remember that the natural logarithm \( \ln(x) \) is used to solve for time \( t \) when the equation involves an exponential decay function.

Learn how to calculate the decay of strontium-90 over time with an initial amount of 400 grams. Find out how much will be left after 10 years and when the amount will reduce to 140 grams using the exponential decay formula and natural logarithm. Suitable for students in Grades 10-12.

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