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:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Radioactive Decay",
"Half-Life Formula"
],
"current_concepts": [
"Exponential Functions",
"Exponential Decay",
"Radioactive Decay",
"Half-Life Formula"
]
}
</pre_analysis>

<reasoning>

Identify given parameters

\[

$$\begin{aligned} &f(t) = m(0.5)^{\frac{t}{h}}\\ &m = 200\text{ mg}\\ &h = 2.7\text{ years} \end{aligned}$$

\]

Formulate the decay equation

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

Calculate remaining mass at t = 12

\[

$$\begin{aligned} &f(12) = 200(0.5)^{\frac{12}{2.7}}\\ &\frac{12}{2.7} \approx 4.444\\ &f(12) \approx 200(0.5)^{4.444} \approx 200(0.0459) \approx 9.2\text{ mg} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(t) = 2.7(0.5)^{\frac{t}{200}}; 2.6\text{ mg}\)</mcq-option>
<mcq-option>(B) \(f(t) = 2.7(0.5)^{t}; 0.0007\text{ mg}\)</mcq-option>
<mcq-option>(C) \(f(t) = 200(0.5)^{t}; 0.05\text{ mg}\)</mcq-option>
<mcq-correct>(D) \(f(t) = 200(0.5)^{\frac{t}{2.7}}; 9.2\text{ mg}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Half-Life Formula"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Radioactive Decay",
"Half-Life Formula"
],
"current_concepts": [
"Exponential Functions",
"Exponential Decay",
"Radioactive Decay",
"Half-Life Formula"
]
}
</pre_analysis>

<reasoning>

Identify given parameters

\[

$$\begin{aligned} &f(t) = m(0.5)^{\frac{t}{h}}\\ &m = 200\text{ mg}\\ &h = 2.7\text{ years} \end{aligned}$$

\]

Formulate the decay equation

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

Calculate remaining mass at t = 12

\[

$$\begin{aligned} &f(12) = 200(0.5)^{\frac{12}{2.7}}\\ &\frac{12}{2.7} \approx 4.444\\ &f(12) \approx 200(0.5)^{4.444} \approx 200(0.0459) \approx 9.2\text{ mg} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(t) = 2.7(0.5)^{\frac{t}{200}}; 2.6\text{ mg}\)</mcq-option>
<mcq-option>(B) \(f(t) = 2.7(0.5)^{t}; 0.0007\text{ mg}\)</mcq-option>
<mcq-option>(C) \(f(t) = 200(0.5)^{t}; 0.05\text{ mg}\)</mcq-option>
<mcq-correct>(D) \(f(t) = 200(0.5)^{\frac{t}{2.7}}; 9.2\text{ mg}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Half-Life Formula"
]
}
</post_analysis>