\(\mathrm{R(t) = 24(0.4)^{(t/3)}}\)The function R models the rate of a chemical reaction, in molecules per second, t minutes after the...
GMAT Advanced Math : (Adv_Math) Questions
The function R models the rate of a chemical reaction, in molecules per second, t minutes after the reaction began. According to this model, what was the reaction rate when the reaction began?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{R(t) = 24(0.4)^{(t/3)}}\) models reaction rate in molecules per second
- \(\mathrm{t}\) represents minutes after the reaction began
- Need to find reaction rate "when the reaction began"
- What this tells us: "When the reaction began" means \(\mathrm{t = 0}\) (zero minutes after start)
2. SIMPLIFY by substitution
- Substitute \(\mathrm{t = 0}\) into the function:
\(\mathrm{R(0) = 24(0.4)^{(0/3)}}\)
- Simplify the exponent:
\(\mathrm{R(0) = 24(0.4)^0}\)
- Apply zero exponent rule - any number to the power 0 equals 1:
\(\mathrm{R(0) = 24(1) = 24}\)
Answer: C. 24
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students don't remember or incorrectly apply the zero exponent rule. They might think \(\mathrm{(0.4)^0 = 0}\) instead of 1, leading to \(\mathrm{R(0) = 24(0) = 0}\). Since 0 isn't among the answer choices, this leads to confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students might misinterpret "when the reaction began" and use a different value for t, such as \(\mathrm{t = 1}\) or confuse the time units. This could lead them to calculate \(\mathrm{R(1) = 24(0.4)^{(1/3)} ≈ 24(0.737) ≈ 17.7}\), causing them to guess among the available choices.
The Bottom Line:
This problem tests whether students can connect everyday language to mathematical notation and apply a fundamental exponent rule. The key insight is recognizing that "when something began" corresponds to time zero in mathematical modeling.