a bacteria population doubles every eight minutes. if the population begins with one cell how long will it take to grow to 512 cells?
well, 512 = 2^9, so it has doubled 9 times...
To determine how long it will take for a bacteria population to grow to a specific number of cells, we can use the concept of exponential growth.
In this case, the bacteria population is doubling every eight minutes.
Let's break down the problem step by step:
1. Start with one cell.
2. After eight minutes, there will be two cells (1 x 2^1).
3. After the next eight minutes (16 minutes in total), there will be four cells (1 x 2^2).
4. After 24 minutes, there will be eight cells (1 x 2^3).
5. Notice that the exponent (the number to which we raise 2) is the same as the number of intervals of eight minutes that have passed.
6. We can generalize this pattern: after n intervals of eight minutes, the population will be 2^n cells.
Now, let's solve the problem by finding the exponent that corresponds to a population of 512 cells:
1 x 2^n = 512
To isolate n, we can take the logarithm of both sides. Let's use the base 2 logarithm (log2):
log2(1 x 2^n) = log2(512)
n = log2(512)
Using a calculator or math software, we find that log2(512) is equal to 9. Therefore, it will take 9 intervals of eight minutes for the population to grow to 512 cells.
To find the total time, we multiply the number of intervals (9) by the length of each interval (8 minutes):
9 x 8 minutes = 72 minutes
So, it will take 72 minutes for the bacteria population to grow from one cell to 512 cells.