Question

the amount of a radioactive substance remaining after \\(t\\) years is given by the function \\(f(t) = m(0.5)^{\frac{t}{h}}\\), where \\(m\\) is the initial mass and \\(h\\) is the half-life in years. iron has a half-life of 2.7 years. which equation gives the mass of a 200 mg iron sample remaining after \\(t\\) years, and approximately how many milligrams remain after 12 years?

\\(f(t) = 2.7(0.5)^{\frac{t}{200}}\\); 2.6 mg
\\(f(t) = 2.7(0.5)^t\\); 0.0007 mg
\\(f(t) = 200(0.5)^t\\); 0.05 mg
\\(f(t) = 200(0.5)^{\frac{t}{2.7}}\\); 9.2 mg

Explanation:

Identify the given parameters

Using the Exponential Decay Modeling knowledge point

$$
LATEXBLOCK0
$$

Substitute parameters into the function

Using the Exponential Decay Modeling knowledge point

$$ f(t) = 200(0.5)^{\frac{t}{2.7}} $$

Calculate the remaining mass after 12 years

Using the Half-Life Calculation knowledge point

$$
LATEXBLOCK1
$$

Match with the given options

Using the Half-Life Calculation knowledge point

$$
LATEXBLOCK2
$$

This matches the fourth option.

Answer:

• (A) \(f(t) = 2.7(0.5)^{\frac{t}{200}}; 2.6\text{ mg}\)
• (B) \(f(t) = 2.7(0.5)^{t}; 0.0007\text{ mg}\)
• (C) \(f(t) = 200(0.5)^{t}; 0.05\text{ mg}\)
• (D) \(f(t) = 200(0.5)^{\frac{t}{2.7}}; 9.2\text{ mg}\) (Correct answer)