Digital SATMath: AlgebraMedium
A biologist is studying a bacterial culture where the population doubles every 3 hours. If the initial population is 200 bacteria, which equation models the population P after t hours?
- AP = 200(t/3)^2
- BP = 200(2)^(t/3)
- CP = 200(2)^t
- DP = 200(3)^t
Show answer & explanationAnswer & explanation
Correct answer: B. P = 200(2)^(t/3)
For exponential growth that doubles over a period, the general formula is P = P₀(2)^(t/d), where P₀ is the initial population, t is the total time, and d is the doubling period. Here, P₀ = 200 and d = 3 hours, so the equation is P = 200(2)^(t/3).
Why the other options are wrong
- A. This equation is quadratic, not exponential, and uses an incorrect base and power.
- C. This equation would model doubling every 1 hour, not every 3 hours.
- D. This equation uses a base of 3, implying tripling, and incorrectly assumes a 1-hour period.
Exponential Growth (Doubling Period)
Exponential growth with a doubling period describes situations where a quantity increases by a factor of 2 over a fixed time interval.
- Formula: P(t) = P₀(2)^(t/d)
- P₀ is initial amount, t is time, d is doubling period.
- The exponent t/d ensures the doubling factor applies correctly over the period.
Memory trick: Doubling: Start with P₀, multiply by 2, raise to time-over-period!