The phrase “triples every fixed number of hours” signals exponential growth: the same multiplication repeats over equal intervals. Start with the initial population, use the tripling factor as the base, and make the exponent count the intervals that have passed. Divide time by the interval length; using the hours directly would apply the factor too often.
Hints
- Hint 1
In an exponential model, tripling means multiplying by once per interval. Begin with the starting number of cells. If is the number of intervals, how would you write the population using rounds of tripling?
- Hint 2
The exponent counts rounds of tripling, not hours. Each round takes hours, so how many rounds have passed after hours? Divide the elapsed time by the time per round.
Step-by-step
Build the growth model
Step 1Write one factor for each tripling
No Desmos needed. The words give the starting amount, growth factor, and interval directly. An exponential model uses repeated multiplication. The culture starts with cells and triples each interval, so if counts the intervals, its size is . The stays in front because it is the starting population.
- Step 2
Count the five-hour intervals
Each interval lasts hours, so divide the elapsed time by : . At , that gives an exponent of , or exactly one tripling. The exponent counts growth intervals, not individual hours.
- Step 3
Complete the function
Replace with in the growth model: . This models the number of bacteria after hours. Choice B.