Element X is a radioactive isotope such that every 5 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 590 grams, how much of the element would remain after 23 years, to the nearest whole number?

This problem describes a classic example of exponential decay, where the amount of a radioactive substance decreases over time. The mass of the substance halves every 5 years, so the decay can be modeled using the following exponential decay formula:

\[
m(t) = m_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

Where:
- \(m(t)\) is the remaining mass after time \(t\),
- \(m_0\) is the initial mass (590 grams),
- \(T\) is the half-life of the element (5 years),
- \(t\) is the elapsed time (23 years).

### Step-by-step solution:
1. The initial mass of the sample, \(m_0\), is 590 grams.
2. The half-life, \(T\), is 5 years.
3. The time elapsed, \(t\), is 23 years.

Substituting these values into the formula:

\[
m(23) = 590 \times \left( \frac{1}{2} \right)^{\frac{23}{5}}
\]

Now, let's calculate the value of \(m(23)\).After 23 years, the remaining mass of Element X would be approximately **24 grams**.

Would you like any further details on this? Here are some related questions to expand on this concept:

1. How would the remaining mass change if the half-life were 10 years instead of 5?
2. What is the general formula for the mass of a radioactive substance after \( n \) half-lives?
3. How long would it take for the mass to decay to 1 gram?
4. How can this exponential decay model be used in fields like carbon dating?
5. What happens to the mass of a substance if the time is less than the half-life?

**Tip:** Radioactive decay follows exponential patterns, which are common in nature and also appear in population growth and financial interest models.

Learn how to calculate the remaining mass of a radioactive isotope, Element X, after 23 years, given that it has a half-life of 5 years. This problem demonstrates exponential decay concepts and half-life calculations, suitable for high school students (Grades 9-12).

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