The amount of a radioactive substance is modeled by the function \(\mathrm{A(t)} = 500\left(\frac{1}{2}\right)^{\frac{t}{120}}\), where \(\mathrm{A(t)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{A(t) = 500(1/2)^{(t/120)}}\) models radioactive substance
  • Need time when amount = half of initial amount

• This tells us we need to find the initial amount first, then set up an equation

2. INFER what the initial amount is

• At \(\mathrm{t = 0}\) (initially): \(\mathrm{A(0) = 500(1/2)^{(0/120)}}\)

\(\mathrm{= 500(1/2)^0}\)

\(\mathrm{= 500(1)}\)

\(\mathrm{= 500}\) grams

• Half of initial amount \(\mathrm{= 500 ÷ 2 = 250}\) grams

3. TRANSLATE "half the initial amount" into an equation

• We need: \(\mathrm{A(t) = 250}\)
• This gives us: \(\mathrm{250 = 500(1/2)^{(t/120)}}\)

4. SIMPLIFY the exponential equation

• Divide both sides by 500: \(\mathrm{250/500 = (1/2)^{(t/120)}}\)
• This gives us: \(\mathrm{1/2 = (1/2)^{(t/120)}}\)

5. INFER the solution strategy

• Since both sides have the same base (1/2), their exponents must be equal
• Left side: \(\mathrm{(1/2)^1}\), so exponent is 1
• Right side: \(\mathrm{(1/2)^{(t/120)}}\), so exponent is \(\mathrm{t/120}\)
• Therefore: \(\mathrm{1 = t/120}\)

6. SIMPLIFY to find the final answer

• Multiply both sides by 120: \(\mathrm{t = 120}\) hours

Answer: 120

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what "half of initial amount" means, thinking they need to find when \(\mathrm{A(t) = 1/2}\) instead of when \(\mathrm{A(t) = 250}\).

They set up: \(\mathrm{1/2 = 500(1/2)^{(t/120)}}\), which leads to incorrect algebraic manipulation. This confusion about the target value causes them to get stuck and abandon systematic solution, often leading to guessing.

Second Most Common Error:

Missing conceptual knowledge about exponent properties: Students don't recognize that when bases are equal in exponential equations, exponents must be equal.

Instead, they try to solve \(\mathrm{1/2 = (1/2)^{(t/120)}}\) using logarithms or other complex methods, making calculation errors or getting overwhelmed by unnecessary complexity. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can translate real-world language into precise mathematical relationships and then apply basic exponential equation solving techniques. The key insight is recognizing that "half-life" problems often simplify elegantly when you set up the equation correctly.