To model the radioactive decay of the substance, we start with the given formula:

$M(t) = M_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$

where:

• $M(t)$ is the mass remaining after $t$ years,
• $M_0$ is the initial mass,
• $h$ is the half-life of the substance,
• $t$ is the time in years.

Given:

• The initial mass $M_0 = 200$ grams,
• The half-life $h = 5$ years.

Substituting $M_0 = 200$ and $h = 5$ into the formula, we get:

$M(t) = 200 \left(\frac{1}{2}\right)^{\frac{t}{5}}$

Thus, the model function that predicts the mass remaining after $t$ years is:

$M(t) = 200 \times \left(\frac{1}{2}\right)^{\frac{t}{5}}$

This function can now be used to calculate the remaining mass for any time $t$ in years.

Would you like to calculate specific values, or do you have any other questions?

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1. What is the remaining mass after 10 years?
2. How would you determine the time required for the substance to decay to 25 grams?
3. Can this formula be adapted for substances with different half-lives?
4. How does the half-life affect the rate of decay in the function?
5. What would the function look like if the initial mass were different?

Tip: The half-life decay formula is exponential, meaning the quantity decreases by a constant percentage over equal time intervals. This property can simplify long-term predictions without recalculating each intermediate step.