Calculating Boat Speed and Travel Time in Still Water

This problem involves finding the time it takes for a boat to travel a specific distance in still water, given its speeds with and against the current. We need to determine the boat's speed in still water first.

Understanding the Given Information

• Speed of Rajiv's boat along the current (downstream speed) = 8 km/hour.
• Speed of Rajiv's boat against the current (upstream speed) = 6 km/hour.
• Distance to travel in still water = 28 km.

Formulating Equations for Speed

Let's define the speeds:

• Let $S_b$ represent the speed of the boat in still water (in km/hour).
• Let $S_c$ represent the speed of the current (in km/hour).

We can express the downstream and upstream speeds using these variables:

• Downstream Speed ($S_b + S_c$) = 8 km/hour
• Upstream Speed ($S_b - S_c$) = 6 km/hour

Step-by-Step Calculation of Boat Speed ($S_b$)

We have a system of two linear equations:

1. $S_b + S_c = 8$
2. $S_b - S_c = 6$

To find the speed of the boat in still water ($S_b$), we can add the two equations together:

$$ (S_b + S_c) + (S_b - S_c) = 8 + 6 $$

Combining like terms:

$$ 2 S_b = 14 $$

Now, solve for $S_b$ by dividing both sides by 2:

$$ S_b = \frac{14}{2} $$

$$ S_b = 7 $$

Therefore, the speed of Rajiv's boat in still water is 7 km/hour.

Calculating the Time Taken in Still Water

The question asks for the time taken to travel 28 km in still water. We use the formula:

$$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$

Using the distance and the calculated speed of the boat in still water:

$$ \text{Time} = \frac{28 \text{ km}}{7 \text{ km/hour}} $$

$$ \text{Time} = 4 \text{ hours} $$

Conclusion

The time taken by Rajiv's boat to sail 28 km in still water is 4 hours.