Digital SATMath: Advanced MathHard
A scientist is observing the population growth of a certain bacteria culture. The population P(t) after t hours is modeled by the function P(t) = P_0 * e^(kt). If the population doubles every 3 hours, what is the value of the growth rate constant k?
- Aln(2) / 3
- Bln(3) / 2
- C2 / 3
- D3 / ln(2)
Show answer & explanationAnswer & explanation
Correct answer: A. ln(2) / 3
The population doubles every 3 hours means that when t=3, P(3) = 2 * P_0. Substitute this into the formula: 2 * P_0 = P_0 * e^(k*3). Divide by P_0: 2 = e^(3k). Take the natural logarithm of both sides: ln(2) = ln(e^(3k)). This simplifies to ln(2) = 3k. Finally, solve for k: k = ln(2) / 3.
Why the other options are wrong
- B. This is a misapplication of the natural logarithm with the wrong numbers.
- C. This incorrectly uses 2 and 3 without the natural logarithm.
- D. This is an inversion of the correct formula.
Exponential Growth Rate Constant
For continuous exponential growth modeled by P(t) = P_0 * e^(kt), the growth rate constant 'k' determines the speed of growth. If given doubling time (T_d), k = ln(2) / T_d.
- e^(kt) is the continuous growth factor.
- Doubling time (T_d) means P(T_d) = 2 * P_0.
- k and T_d are inversely related through ln(2).
Memory trick: Doubling Time T, K is Ln 2 Over T!