ACT (Enhanced)MathematicsEasy
A biologist is observing a population of fruit flies. The population, P(t), after t days can be modeled by the equation P(t) = 500 / (1 + 4e^(-0.2t)). What is the initial population of fruit flies?
- A500
- B100
- C200
- D50
Show answer & explanationAnswer & explanation
Correct answer: B. 100
To find the initial population, substitute t=0 into the given logistic growth equation. This simplifies the exponential term, allowing direct calculation of P(0).
Why the other options are wrong
- A. Incorrect. This is the carrying capacity, not the initial population.
- C. Incorrect. This results from an arithmetic error, possibly miscalculating 4e^0.
- D. Incorrect. This results from an arithmetic error, likely dividing by 10 instead of 5.
Initial Value of Logistic Function
The initial value of a logistic function P(t) = K / (1 + Ae^(-kt)) is found by setting t=0, which simplifies the exponential term to 1.
- Set t=0 to find the initial value.
- e^0 = 1, simplifying the calculation.
- The initial value is always K / (1 + A) for the standard form.
Memory trick: Start at 'Zero Time' to see the 'Initial Climb'.