# If you start a biology experiment with 5,000,000 cells and 25% of the cells are dying every minute,how long will it be before there are fewer than 1000 cells?

## 5000000(3/4)^t < 1000

(3/4)^t < 1/5000
t ln(3/4) < ln(1/5000)
t > 29.6 minutes

## To determine how long it will take for there to be fewer than 1000 cells, we need to calculate the number of minutes it takes for the population to decrease to that level.

2. If 25% of the cells die every minute, then the number of surviving cells after each minute can be calculated by subtracting 25% (or 0.25) of the previous count:
- After 1 minute: 5,000,000 - (0.25 * 5,000,000) = 3,750,000 cells
- After 2 minutes: 3,750,000 - (0.25 * 3,750,000) = 2,812,500 cells
- After 3 minutes: 2,812,500 - (0.25 * 2,812,500) = 2,109,375 cells
- And so on...
3. Continue this calculation until the number of cells falls below 1000.
- After 4 minutes: 1,582,031 cells
- After 5 minutes: 1,186,523 cells
- After 6 minutes: 889,892 cells
- After 7 minutes: 667,419 cells
- After 8 minutes: 500,564 cells
- After 9 minutes: 375,423 cells
- After 10 minutes: 281,567 cells
- After 11 minutes: 211,175 cells
- After 12 minutes: 158,382 cells
- After 13 minutes: 118,786 cells
- After 14 minutes: 89,090 cells
- After 15 minutes: 66,817 cells
- After 16 minutes: 50,114 cells
- After 17 minutes: 37,585 cells
- After 18 minutes: 28,189 cells
- After 19 minutes: 21,142 cells
- After 20 minutes: 15,857 cells
- After 21 minutes: 11,892 cells
- After 22 minutes: 8,919 cells
- After 23 minutes: 6,689 cells
- After 24 minutes: 5,017 cells

Therefore, it will be around 24 minutes before there are fewer than 1000 cells remaining in the population.

## To answer this question, we need to calculate the number of cells remaining after each minute until the count reaches fewer than 1000 cells.

Initially, we have 5,000,000 cells. After one minute, 25% of the cells die, which means 75% of the cells remain. So, the number of cells after the first minute would be 5,000,000 * 0.75.

We can continue this process iteratively until the number of cells drops below 1000. Here's a step-by-step breakdown:

1. After 1 minute: 5,000,000 * 0.75 = 3,750,000 cells remaining.
2. After 2 minutes: 3,750,000 * 0.75 = 2,812,500 cells remaining.
3. After 3 minutes: 2,812,500 * 0.75 = 2,109,375 cells remaining.

We can continue this process until the number of cells falls below 1000.

Let's calculate how many minutes it takes for the number of cells to drop below 1000:

4. After 4 minutes: 2,109,375 * 0.75 = 1,582,031 cells remaining.
5. After 5 minutes: 1,582,031 * 0.75 = 1,186,523.25 cells remaining (assuming partial cells exist, but we round down).
6. After 6 minutes: 1,186,523.25 * 0.75 = 889,892.4375 cells remaining.
7. After 7 minutes: 889,892.4375 * 0.75 = 667,419.328125 cells remaining.
8. After 8 minutes: 667,419.328125 * 0.75 = 500,564.49609375 cells remaining.

After 8 minutes, there are approximately 500,564 cells remaining, which is fewer than 1000 cells. Therefore, it takes 8 minutes for the cell count to drop below 1000.