Problem Statement: A bacterial cell divides every one minute. It takes 15 minutes for a cup to be one fourth full. How much time will it take to fill the cup?

Solution:

To solve this problem, we need to use the concept of exponential growth.

Exponential Growth: Exponential growth is a pattern of growth where the growth rate is proportional to the current size of the population. In other words, the rate at which the population grows increases as the population gets larger.

Calculating the Growth Rate: If a bacterial cell divides every one minute, then the growth rate is 2 (i.e. one cell becomes two cells every minute).

Calculating the Number of Cells: In order to calculate the number of cells, we need to use the formula:

N = N0 x 2^t

where N is the final number of cells, N0 is the initial number of cells, and t is the time in minutes.

Calculating the Number of Cells in 15 Minutes: If it takes 15 minutes for a cup to be one fourth full, then the cup is one eighth full in 14 minutes (since the bacterial growth rate is exponential). Therefore, the number of cells in the cup after 15 minutes is:

N = N0 x 2^15

where N0 is the initial number of cells (which is one), and t is 15 minutes.

N = 1 x 2^15
N = 32,768

Calculating the Time to Fill the Cup: To calculate the time it takes to fill the cup, we need to use the formula:

t = log2(N/N0)

where t is the time in minutes, N is the final number of cells, and N0 is the initial number of cells.

Using the values we have calculated earlier, we can calculate the time it takes to fill the cup:

t = log2(32768/1)
t = 15

However, we need to add the additional minute it takes for the cup to be one eighth full, so the total time it takes to fill the cup is:

t = 15 + 1
t = 16

Therefore, it takes 16 minutes for the cup to be full.

Let's assume for simplification, the cup can hold total 100 bacteria.In fifteen minutes, it is 1/4th filled, means 25 bacteria are present.Now each bacterial cell divides in 1 minute. So, each of 25 bacterial cells will divide and after 1 minute will form 25 more cells. Total no of bacteria is 50.Similarly each of the 50 bacteria will divide & after 1 minute will form another 50 cells. Total no of bacteria is now 100.The cup is full. Time taken = 15+1+1 = 17 min