The half-life of phosphorus-32 is approximately 24.3
 days.

Step 2 of 3 :  How much of a 3
 gram sample of phosphorus-32 would remain after 9
 days? Round to three decimal places.

To solve this problem, we can use the formula for exponential decay related to half-life:

\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
\]

Where:
- \( N(t) \) is the amount of substance remaining after time \( t \).
- \( N_0 \) is the initial amount of substance.
- \( T_{1/2} \) is the half-life of the substance.
- \( t \) is the elapsed time.

Given:
- \( N_0 = 3 \) grams (initial amount)
- \( T_{1/2} = 24.3 \) days (half-life of phosphorus-32)
- \( t = 9 \) days (elapsed time)

Plugging the values into the formula:

\[
N(9) = 3 \times \left(\frac{1}{2}\right)^{\frac{9}{24.3}}
\]

First, calculate the exponent:

\[
\frac{9}{24.3} \approx 0.370
\]

Now calculate the remaining amount:

\[
N(9) = 3 \times \left(\frac{1}{2}\right)^{0.370}
\]

\[
N(9) = 3 \times 0.766 \approx 2.298
\]

So, approximately **2.298 grams** of the phosphorus-32 would remain after 9 days.

Would you like more details on this calculation or have any other questions?

Here are some related questions to consider:
1. How would the remaining amount change if the half-life was longer or shorter?
2. How much phosphorus-32 would remain after one full half-life period (24.3 days)?
3. What would be the remaining amount after 50 days?
4. How is half-life used to date ancient objects using carbon dating?
5. What are some practical applications of knowing the half-life of a substance?

**Tip:** Remember, the concept of half-life applies not only to radioactive decay but also to many other processes, such as drug metabolism and certain chemical reactions.

Learn how to calculate the remaining amount of phosphorus-32 after 9 days using the exponential decay formula. Given an initial sample of 3 grams and a half-life of 24.3 days, determine the remaining quantity with precision. Ideal for high school students studying exponential decay and radioactive substances.

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