In this ques. you have to look from last i.e the cup get completely filled in 60 min, just 3 min earlier( at 57 min) the cup must be 1/2 filled ( As the no. of bacteria becomes double in 3 min) , just 3 min earlier of half filled cup i.e at 54 min the cup must be 1/4th fill with bacteria..............In other words ...........In 54 min cup 1/4th...in next 3min cup 1/2 .....in next 3 min cup FULL....

So ans is 54 min

Given Information:
- Each bacterial cell divides in every 3 minutes.
- The cup completely fills in one hour.

To find:
- The time taken to fill 1/4th part of the cup.

Let's break down the problem into smaller parts to find a solution.

1. Calculate the number of bacterial cells at the end of one hour:

- Since each bacterial cell divides every 3 minutes, there will be 20 divisions in one hour (60 minutes / 3 minutes per division = 20 divisions).
- Each division doubles the number of bacterial cells.
- Therefore, the number of bacterial cells at the end of one hour will be 2^20 = 1,048,576.

2. Calculate the volume of the cup:

- Since the cup completely fills in one hour, we can assume that the volume of the cup remains constant throughout.
- Let's assume the cup has a volume of V.

3. Calculate the volume filled by each bacterial cell:

- Since each bacterial cell divides, it will generate two daughter cells.
- Therefore, the volume filled by each bacterial cell is V/2.

4. Calculate the volume filled by all bacterial cells at the end of one hour:

- The total number of bacterial cells at the end of one hour is 1,048,576.
- Therefore, the total volume filled by all bacterial cells is 1,048,576 * (V/2) = 524,288V.

5. Calculate the volume filled by 1/4th part of the cup:

- Since we need to find the time taken to fill 1/4th part of the cup, the volume filled by 1/4th part of the cup is (1/4) * V.

Now, let's determine the time taken to fill 1/4th part of the cup.

- For the cup to fill, the volume filled by all bacterial cells (524,288V) should be equal to the volume of the cup (V).
- Therefore, we can set up the equation:
524,288V = V/4
524,288 = 1/4
V = 524,288 * 4 = 2,097,152

- The time taken to fill 1/4th part of the cup is the time taken for the bacterial cells to reach a volume of 1/4th of the cup's volume, which is 2,097,152.
- Since each bacterial cell divides every 3 minutes, the time taken will be 20 divisions * 3 minutes per division = 60 minutes.

Therefore, the time taken to fill 1/4th part of the cup is 60 minutes, which corresponds to option 1) 58 minutes.