Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A biologist is studying a population of bacteria that doubles every 30 minutes. If the initial population is 100 bacteria, which equation models the population P(t) after 't' hours?
- AP(t) = 100 * 2^(t)
- BP(t) = 100 * (1 + 0.5t)
- CP(t) = 100 * 2^(t/30)
- DP(t) = 100 * 2^(2t)
Show answer & explanationAnswer & explanation
Correct answer: D. P(t) = 100 * 2^(2t)
The population doubles every 30 minutes. Since the time 't' is in hours, we need to convert the doubling period to hours: 30 minutes = 0.5 hours. The number of doubling periods in 't' hours is t / 0.5, which simplifies to 2t. Therefore, the equation is P(t) = 100 * 2^(2t).
Why the other options are wrong
- A. This implies doubling occurs every hour, not every 30 minutes.
- B. This is a linear growth model, not exponential doubling.
- C. Incorrectly uses 't/30' as the exponent directly without converting units or understanding the doubling periods per hour.
Exponential Growth with Different Time Units
Modeling exponential growth when the growth period is different from the unit of time used for the independent variable.
- The exponent must represent the number of growth periods that have occurred.
- If the growth period is 'k' units and the time variable is 't' in those units, the exponent is t/k.
- Ensure consistency in time units (e.g., if growth is every 30 min, and 't' is in hours, convert 30 min to 0.5 hours, so the exponent is t/0.5 or 2t).
Memory trick: Match the units! Doubling period must fit the time variable!