A single cell divides in two every 5 minutes and each new cell continues to divide every 5 minutes. How long does it take for the cell population to reach at least 1 million?
The cells divide every 5 minutes, starting from 1. Let x equal every 5 minute period. Then:
1,000,000=1*2^x
ln 1,000,000=ln 2^x=x ln 2
x=19.93
19.93 x 5=99.66 minutes before the cell population reaches 1,000,000.
unowen is correct. I did not read the problem carefully.
Well, if each cell divides into two every 5 minutes, we can assume exponential growth. Let me put on my clown nose and do some calculations for you!
After 5 minutes, we have 2 cells. After 10 minutes, we have 4 cells. After 15 minutes, we have 8 cells. Do you see where I'm going with this?
Using the power of multiplication, we can determine that after 25 minutes, we have approximately 1,048,576 cells. Aha! We're beyond the 1 million mark!
So, it takes approximately 25 minutes for the cell population to reach at least 1 million. Isn't biology multiplying fun?
To solve this problem, we need to find out how many divisions are required for the cell population to reach at least 1 million.
Let's break down the problem step by step:
1. Start with 1 cell.
2. After 5 minutes, the cell divides into 2 cells.
3. After another 5 minutes (total 10 minutes), each of the 2 cells divides into 2 new cells, resulting in a total of 4 cells.
4. After another 5 minutes (total 15 minutes), each of the 4 cells divides into 2 new cells, resulting in a total of 8 cells.
You can see the pattern here: The number of cells doubles after every 5 minutes.
At this point, we can calculate how many divisions are needed to reach at least 1 million cells:
- 1 cell initially
- 2 cells after 5 minutes
- 4 cells after 10 minutes
- 8 cells after 15 minutes
If we continue this pattern, we can see that after each subsequent 5-minute interval, the number of cells doubles again:
- 16 cells after 20 minutes
- 32 cells after 25 minutes
- 64 cells after 30 minutes
We can observe that after every 5 minutes, the number of cells doubles. Therefore, after n intervals of 5 minutes, the number of cells will be 2^n.
We need to find the smallest value of n such that 2^n is equal to or greater than 1 million.
2^n ≥ 1,000,000
Taking the log base 2 of both sides of the inequality:
n ≥ log2(1,000,000)
Using a calculator, we find that log2(1,000,000) is approximately equal to 19.93.
Since n represents the number of 5-minute intervals, we need to round up to the nearest whole number, which is 20.
Therefore, it will take at least 20 * 5 = 100 minutes for the cell population to reach at least 1 million cells.