A population that doubles after a fixed interval follows an exponential model: the starting amount is multiplied by once per interval. Divide the elapsed time by the interval length to count how many times that multiplication applies. Using hours directly as the exponent would make the population double every hour, rather than every stated interval.
Hints
- Hint 1
In an exponential model, the number in front is the starting population. At time , the exponent is , and a growth factor to the zero power is . What number belongs in front?
- Hint 2
The exponent counts doubling intervals, not individual hours. Divide the hours elapsed by the hours in one doubling interval. What should the exponent be after one full interval?
Step-by-step
Build the model from the start, factor, and interval
Step 1Set the starting population
No Desmos needed. The model comes directly from the starting amount and doubling interval. In an exponential model, the number in front is the population at time , because any growth factor to the power is . So 800 goes in front: , where is the factor for one interval and counts intervals.
- Step 2
Use the doubling factor
To double means to multiply the current population by . So the growth factor is : .
- Step 3
Count the six-hour intervals
One interval is a six-hour stretch. Divide the elapsed hours by to count those stretches: . At , that count is , so the factor applies once. The exponent counts six-hour intervals, not individual hours.
- Step 4
Write the population model
Replace with : . This gives 800 bacteria at the start and doubles the population every six hours. Choice B.