ACT (Enhanced)MathematicsHard
A scientist is studying a chemical reaction where the concentration of a product, C(t), in moles per liter, at time t hours, is given by C(t) = 5 - 4e^(-0.2t). What is the maximum possible concentration of the product as time approaches infinity?
- A9 mol/L
- B5 mol/L
- C0.2 mol/L
- D4 mol/L
Show answer & explanationAnswer & explanation
Correct answer: B. 5 mol/L
As time (t) approaches infinity, the exponential term e^(-0.2t) approaches 0. Therefore, C(t) approaches 5 - 4(0), which simplifies to 5.
Why the other options are wrong
- A. This would be the sum of 5 and 4, which is incorrect as the exponential term approaches zero.
- C. This is the decay rate constant, not the maximum concentration.
- D. This is the coefficient of the exponential term, which is subtracted, not the limit.
Limit of Exponential Decay (Asymptote)
The value an exponential decay function approaches as its independent variable tends towards infinity, often representing a carrying capacity or saturation point.
- For functions like y = A - Be^(-kt), as t → ∞, e^(-kt) → 0.
- The limit is then A, representing a horizontal asymptote.
- This value is the maximum or minimum the function can reach.
Memory trick: Time to infinity, e to the negative becomes nought, the constant's the limit, a valuable thought!