The distance between Dakhhineshwar and Babughat is 90km. A boat while going downstream takes 3 hours to complete this distance. Bulbul a passenger in a boat dropped a polythene when she was at Dakhhineshwar. After reaching Babughat, she realized that the polythene was dropped. She immediately took the boat and started going upstream. After how much time (in hours), after the polythene was dropped, will she be able to find the polythene if the speed of the boat is 20 kmph in still water.

To determine how much time after the polythene was dropped Bulbul will be able to find it, let's analyze the situation step-by-step:

1. **Calculate the speed of the boat downstream and upstream:**
   - The distance between Dakhineshwar and Babughat is 90 km.
   - The boat takes 3 hours to travel downstream, so the downstream speed \(v_{\text{downstream}}\) is:
     \[
     v_{\text{downstream}} = \frac{\text{distance}}{\text{time}} = \frac{90 \text{ km}}{3 \text{ hours}} = 30 \text{ kmph}
     \]

2. **Determine the speed of the current:**
   - Let \(v_{\text{boat}}\) be the speed of the boat in still water and \(v_{\text{current}}\) be the speed of the current.
   - Given \(v_{\text{boat}} = 20 \text{ kmph}\),
   - Downstream speed: \(v_{\text{downstream}} = v_{\text{boat}} + v_{\text{current}} = 30 \text{ kmph}\)
   - Therefore, \(v_{\text{current}} = v_{\text{downstream}} - v_{\text{boat}} = 30 \text{ kmph} - 20 \text{ kmph} = 10 \text{ kmph}\).

3. **Calculate the upstream speed:**
   - Upstream speed: \(v_{\text{upstream}} = v_{\text{boat}} - v_{\text{current}} = 20 \text{ kmph} - 10 \text{ kmph} = 10 \text{ kmph}\)

4. **Determine the time taken to realize the polythene is dropped and start the return journey:**
   - The time taken to travel from Dakhineshwar to Babughat downstream is 3 hours.
   - During this time, the polythene is drifting with the current at 10 kmph.
   - Distance covered by the polythene in 3 hours:
     \[
     \text{Distance} = \text{speed} \times \text{time} = 10 \text{ kmph} \times 3 \text{ hours} = 30 \text{ km}
     \]

5. **Calculate the time taken to catch up with the polythene:**
   - Let \(t\) be the time taken by the boat to catch up with the polythene after it starts the return journey.
   - The boat's upstream speed relative to the polythene's speed is:
     \[
     v_{\text{relative}} = v_{\text{upstream}} + v_{\text{current}} = 10 \text{ kmph} + 10 \text{ kmph} = 20 \text{ kmph}
     \]
   - Distance to be covered to catch the polythene is 30 km.
   - Time taken to cover this distance at the relative speed:
     \[
     t = \frac{\text{distance}}{v_{\text{relative}}} = \frac{30 \text{ km}}{20 \text{ kmph}} = 1.5 \text{ hours}
     \]

6. **Total time after the polythene was dropped until it is found:**
   - The total time \(T\) is the sum of the time taken to reach Babughat and the time taken to catch the polythene:
     \[
     T = 3 \text{ hours} + 1.5 \text{ hours} = 4.5 \text{ hours}
     \]

Therefore, Bulbul will find the polythene 4.5 hours after it was dropped.

Would you like any further details or have any other questions?

Here are some other questions you might want to ask:
1. How to find the speed of a boat in still water?
2. How does the speed of the current affect the boat's speed?
3. How to calculate relative speed in different scenarios?
4. How to solve problems involving distances and relative speeds?
5. How to determine time and distance in upstream and downstream problems?

**Tip:** When solving problems involving relative speed, always account for the directions and magnitudes of the speeds involved.

Learn how to calculate the time it takes to find a dropped polythene in a river using relative speed and distance calculations. This problem involves determining the downstream and upstream speeds of a boat and the speed of the current in a river scenario. Suitable for advanced high school students and above.

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