A culture that doubles at fixed intervals follows an exponential model: starting amount times a factor raised to the number of intervals. Divide elapsed time by the length of one interval to find the exponent, then set the model equal to the target. The tempting slip is to multiply time by the interval length, which counts far too many doublings.
Hints
- Hint 1
In an exponential model, the starting amount goes in front, and the base is the factor applied each interval. What number represents the starting count, and what factor represents one doubling?
- Hint 2
The exponent counts six-hour intervals, not individual hours. After hours, it should equal because the culture has doubled once. What expression in gives you that count?
- Hint 3
To find when a model reaches a target, set the cell count after hours equal to the target count. Keep the starting count inside the model rather than swapping it with the target.
Step-by-step
Count the doubling intervals
Step 1Model the repeated doubling
No Desmos needed. Build the equation; you don't need to solve for . Let count the six-hour intervals. Starting at cells and multiplying by each interval gives an exponential model: , where is the cell count.
- Step 2
Count intervals from hours
Divide the elapsed hours by the hours in one interval: . At , this gives one interval, as it should. The exponent counts intervals, not hours.
- Step 3
Write the count in terms of time
Replace with in the cell-count model: . Here means the number of cells after hours.
- Step 4
Set the count equal to the target
The question asks when the culture reaches the target count of , so set the model equal to it: . This equation determines the time in hours. Choice B.