Let's go step by step and solve the problem.

Given information:

• Initial amount of radioactive isotope: 5 kg
• Half-life: 12 years
• The decay follows an exponential decay model:

$g(x) = A \cdot \left(\frac{1}{2}\right)^{\frac{x}{h}}$

where:

• $A = 5$ kg (initial amount),
• $h = 12$ years (half-life),
• $x$ is the number of years.

(a) Writing the function $g(x)$:

$g(x) = 5 \cdot \left(\frac{1}{2}\right)^{\frac{x}{12}}$

(b) Finding how much is left after 12 years:

$g(12) = 5 \cdot \left(\frac{1}{2}\right)^{\frac{12}{12}}$

$g(12) = 5 \cdot \frac{1}{2}$

$g(12) = 2.5 \text{ kg}$

(c) Finding how much is left after 6 years:

$g(6) = 5 \cdot \left(\frac{1}{2}\right)^{\frac{6}{12}}$

$g(6) = 5 \cdot \left(\frac{1}{2}\right)^{\frac{1}{2}}$

Approximating $\left(\frac{1}{2}\right)^{0.5} \approx 0.707$:

$g(6) \approx 5 \times 0.707$

$g(6) \approx 3.54 \text{ kg}$

Final Answers:

• Function: $g(x) = 5 \cdot \left(\frac{1}{2}\right)^{\frac{x}{12}}$
• Amount left after 12 years: 2.5 kg
• Amount left after 6 years: 3.54 kg

Would you like a more detailed explanation? Here are some related questions:

1. How does the half-life formula relate to exponential decay?
2. What is the general form of an exponential decay function?
3. How do we calculate the remaining mass for a different half-life?
4. What happens to the amount of isotope after multiple half-lives?
5. How can this model be used in real-world applications like carbon dating?

Tip: The half-life formula is useful in many scientific fields, including chemistry, physics, and environmental science! 🚀