A laboratory begins an experiment with 400.0 grams of a radioactive isotope. The isotope decays exponentially, and no additional material...

1. TRANSLATE the problem information

• Given information:
  • Initial mass at \(\mathrm{t = 0}\): 400.0 grams
  • Mass at \(\mathrm{t = 10}\) hours: 43.0 grams
  • Need exponential decay function \(\mathrm{M(t)}\)
• This tells us we need the form \(\mathrm{M(t) = P \times (decay\ factor)^t}\) where \(\mathrm{P = 400.0}\)

2. INFER the approach

• Since we know two points on the exponential curve, we can find the decay factor
• The condition \(\mathrm{M(10) = 43.0}\) will let us solve for the unknown decay factor
• Once we have the decay factor, we can write the complete function

3. Set up the equation using the given condition

From \(\mathrm{M(t) = 400.0 \times (decay\ factor)^t}\) and \(\mathrm{M(10) = 43.0}\):

\(\mathrm{400.0 \times (decay\ factor)^{10} = 43.0}\)

4. SIMPLIFY to find the decay factor

• Divide both sides by 400.0:

\(\mathrm{(decay\ factor)^{10} = \frac{43.0}{400.0} = 0.1075}\)

• Take the 10th root of both sides (use calculator):

\(\mathrm{decay\ factor = (0.1075)^{\frac{1}{10}} \approx 0.7984 \approx 0.8}\)

5. Write the complete function and verify

• The function is: \(\mathrm{M(t) = 400.0 \times (0.8)^t}\)
• SIMPLIFY to verify:

\(\mathrm{M(10) = 400.0 \times (0.8)^{10}}\)

\(\mathrm{= 400.0 \times (0.1073741824)}\)

\(\mathrm{\approx 43.0}\) ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to use the condition \(\mathrm{M(10) = 43.0}\) to find the specific decay factor. Instead, they try to guess which answer "looks right" based on the initial value of 400.0.

They might notice that choices A, B, and E all start with 400.0 (which is correct for the initial amount), but then randomly pick between the decay factors 0.8, 0.2, or 1.2 without doing the calculation. Since \(\mathrm{1.2 \gt 1}\) represents growth (not decay), they might eliminate E, but still guess between A and B.

This may lead them to select Choice B (\(\mathrm{M(t) = 400.0 \times (0.2)^t}\)) or causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the initial amount should be in the exponential function. They might think the function should start with the remaining amount (43.0) or the amount that decayed (\(\mathrm{357.0 = 400.0 - 43.0}\)).

This leads them to consider choices C or D, not realizing that the exponential decay model always starts with the original amount at \(\mathrm{t = 0}\).

This may lead them to select Choice C (\(\mathrm{M(t) = 357.0 \times (0.8)^t}\)) or Choice D (\(\mathrm{M(t) = 43.0 \times (0.8)^t}\)).

The Bottom Line:

Success requires both understanding the structure of exponential decay functions AND using the given data point to determine the specific decay factor through algebraic manipulation.