The​ half-life of a certain radioactive substance is 

1414 

days. There are 

2.52.5 

grams of the substance initially. Type an expression for the amount of substance as a function of t. When will there be less than 1 g​ remaining?

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Part 1

What is the​ expression?




To express the amount of a radioactive substance as a function of time \( t \), we can use the formula for exponential decay:

\[
A(t) = A_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

Where:
- \( A(t) \) is the amount of the substance at time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( T \) is the half-life of the substance,
- \( t \) is the time elapsed.

### Part 1: Expression for the amount of substance
Given:
- \( A_0 = 2.5 \) grams (initial amount),
- \( T = 1414 \) days (half-life).

The expression for the amount of substance remaining at time \( t \) is:

\[
A(t) = 2.5 \cdot \left( \frac{1}{2} \right)^{\frac{t}{1414}}
\]

### Part 2: When will there be less than 1 gram remaining?
We need to find \( t \) when \( A(t) < 1 \) gram. This means solving the inequality:

\[
2.5 \cdot \left( \frac{1}{2} \right)^{\frac{t}{1414}} < 1
\]

We can solve this inequality to find the value of \( t \).

Let's calculate the time.The time when there will be less than 1 gram of the substance remaining is approximately 1869 days.

Would you like further clarification on the steps or more details? 

Here are some related questions to expand your understanding:
1. What is the significance of the half-life in radioactive decay?
2. How does the amount of a substance change as time progresses in exponential decay?
3. Can you find the amount of substance after 500 days using the given formula?
4. How does changing the initial amount affect the decay rate?
5. What happens when the half-life is doubled?

**Tip:** When solving exponential decay problems, remember that the half-life helps in finding how quickly a substance reduces by half.

The half-life of a certain radioactive substance is 1414 days. There are 2.5 grams of the substance initially. Type an expression for the amount of substance as a function of t. When will there be less than 1 g remaining?

Learn how to solve a radioactive decay problem with a half-life of 1414 days and an initial amount of 2.5 grams. Discover how to express the amount of the substance as a function of time and find when there will be less than 1 gram remaining. Suitable for Grades 9-12.

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