A boat takes `2 hours` to travel `8 km` and back in still water lake.With water velocity of `4 kmph`,the time taken for going upstream of `8km` and coming back is

Options

A. (a)160 minute
B. (b)80 minute
C. (c)100 minute
D. (d)120 minute

Answer

To solve the problem, we need to determine the time taken for a boat to travel upstream and downstream a distance of 8 km in a lake with a current. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the speed of the boat in still water
Given that the boat takes 2 hours to travel 8 km downstream and back, we can find the speed of the boat in still water.

- **Total distance** = 8 km (downstream) + 8 km (upstream) = 16 km
- **Total time** = 2 hours

Using the formula for speed:
\[ \text{Speed of the boat in still water (V)} = \frac{\text{Total distance}}{\text{Total time}} = \frac{16 \text{ km}}{2 \text{ hours}} = 8 \text{ km/h} \]

### Step 2: Identify the speed of the river
The speed of the river (current) is given as:
\[ \text{Speed of the river (Vr)} = 4 \text{ km/h} \]

### Step 3: Calculate the effective speeds for upstream and downstream
- **Downstream speed (Vd)**: When going downstream, the current aids the boat's speed.
\[ V_d = V + Vr = 8 \text{ km/h} + 4 \text{ km/h} = 12 \text{ km/h} \]

- **Upstream speed (Vu)**: When going upstream, the current opposes the boat's speed.
\[ V_u = V - Vr = 8 \text{ km/h} - 4 \text{ km/h} = 4 \text{ km/h} \]

### Step 4: Calculate the time taken for upstream and downstream
- **Time taken downstream (Td)**:
\[ T_d = \frac{\text{Distance}}{\text{Downstream speed}} = \frac{8 \text{ km}}{12 \text{ km/h}} = \frac{2}{3} \text{ hours} \]

- **Time taken upstream (Tu)**:
\[ T_u = \frac{\text{Distance}}{\text{Upstream speed}} = \frac{8 \text{ km}}{4 \text{ km/h}} = 2 \text{ hours} \]

### Step 5: Calculate the total time for the round trip
The total time taken for the trip is the sum of the time taken upstream and downstream:
\[ T_{\text{total}} = T_d + T_u = \frac{2}{3} \text{ hours} + 2 \text{ hours} = \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \text{ hours} \]

### Step 6: Convert total time into minutes
To convert hours into minutes:
\[ T_{\text{total}} = \frac{8}{3} \text{ hours} \times 60 \text{ minutes/hour} = 160 \text{ minutes} \]

### Final Answer
The total time taken for going upstream of 8 km and coming back is **160 minutes**.

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