The mass, A, in grams, of a radioactive sample remaining after h hours due to radioactive decay is given by...

1. TRANSLATE the unit relationship

• Given information:
  • Original equation: \(\mathrm{A = 50(0.98)^h}\) (h in hours)
  • Need equation in terms of d (days)
  • Unit conversion: 1 day = 24 hours

• What this tells us: If d represents days, then \(\mathrm{h = 24d}\)

2. INFER the substitution strategy

• Since we want the equation in terms of d instead of h, we need to replace h with an expression involving d
• The key insight: wherever we see h in the original equation, we substitute 24d

3. TRANSLATE the substitution into the equation

• Original equation: \(\mathrm{A = 50(0.98)^h}\)
• Replace h with 24d: \(\mathrm{A = 50(0.98)^{24d}}\)
• This matches choice (C) exactly

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students reverse the unit relationship, writing \(\mathrm{d = 24h}\) instead of \(\mathrm{h = 24d}\).

When they think "1 day = 24 hours," they mistakenly conclude that the number of days equals 24 times the number of hours. This backwards thinking leads them to substitute d/24 for h, giving \(\mathrm{A = 50(0.98)^{d/24}}\).

This may lead them to select Choice B (\(\mathrm{A = 50(0.98)^{d/24}}\))

Second Most Common Error:

Poor INFER reasoning: Students think they need to modify the base (\(\mathrm{0.98}\)) rather than understanding that time unit changes affect the exponent.

They might attempt to convert \(\mathrm{0.98}\) to a "daily rate" by raising it to some power or dividing by 24, not realizing that the conversion happens through exponent substitution.

This leads to confusion and guessing between choices A and D.

The Bottom Line:

This problem tests whether students can correctly set up unit conversion relationships and understand that changing time variables in exponential functions requires exponent substitution, not base modification.