A bacteria culture contained 500 cells initially and the number of cells tripled every 6 hours. Which equation gives the number of cells in the culture hours after the observation began?
Repeated multiplication at fixed time intervals is exponential growth. Find the starting count, the multiplier for one interval, and how long that interval lasts. Divide elapsed time by the interval length to count the multiplications. The interval length belongs in the exponent, not in the base: the base is the multiplier.
Hints
- Hint 1
The initial value is the count when no time has passed. In an exponential model, it goes in front of the power. Which given number is the initial count?
- Hint 2
The growth factor is the number multiplying the current count after one interval. A factor greater than grows a population; a factor less than shrinks it. Which direction does tripling describe?
- Hint 3
The exponent counts growth intervals. Each interval lasts hours, so what expression in equals when ?
Step-by-step
Build the exponential model
Step 1Identify the starting count
No Desmos needed. The starting value, factor, and interval come straight from the words. At , no growth interval has passed, so . That makes the initial value, the number in front of the power.
- Step 2
Identify the growth factor
Tripled means multiply the current count by each time. That is the growth factor. Using would make the count smaller instead.
- Step 3
Count the growth intervals
The exponent must count six-hour intervals, not individual hours. In hours, there are such intervals. At , the exponent is , so the culture has tripled once.
- Step 4
Write the cell-count equation
Combine the starting count, factor, and number of intervals to get the number of cells after hours:
Choice C.