Understanding Rashi's Travel Problem

The question asks us to find the speed Rashi needed to travel the remaining distance, given details about her overall journey and specific parts of it.

Here's a breakdown of the information given:

• Total distance Rashi travelled: 400 km
• Overall average speed: 50 km/h
• First leg: First half distance (200 km) covered in 2 hours.
• Second lag: Covered in 1 hour at a speed that is half the speed of the first leg.
• Goal: Find the speed for the remaining distance.

Calculating Rashi's Total Travel Time

First, we need to determine the total time Rashi had for her entire journey. We can use the formula for average speed:

Average Speed = Total Distance / Total Time

Rearranging the formula to find Total Time:

Total Time = Total Distance / Average Speed

Plugging in the values:

$$ \text{Total Time} = \frac{400 \text{ km}}{50 \text{ km/h}} = 8 \text{ hours} $$

So, Rashi had a total of 8 hours for her journey.

Analyzing the First Travel Leg

The problem states Rashi covered the first half of the distance in 2 hours.

• Distance for the first leg: $$ \frac{400 \text{ km}}{2} = 200 \text{ km} $$
• Time taken for the first leg: 2 hours
• Speed during the first leg: $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{200 \text{ km}}{2 \text{ h}} = 100 \text{ km/h} $$

Analyzing the Second Travel Lag

Next, we look at the second lag described:

• Time taken for the second lag: 1 hour
• Speed for the second lag: Half the speed of the first leg.
$$ \text{Speed}_{\text{second}} = \frac{\text{Speed}_{\text{first}}}{2} = \frac{100 \text{ km/h}}{2} = 50 \text{ km/h} $$
• Distance covered during the second lag:
$$ \text{Distance}_{\text{second}} = \text{Speed}_{\text{second}} \times \text{Time}_{\text{second}} = 50 \text{ km/h} \times 1 \text{ h} = 50 \text{ km} $$

Determining Remaining Travel Segments

Now, let's find out how much distance and time are left:

• Total distance covered so far = Distance (Leg 1) + Distance (Lag 2)
$$ \text{Distance Covered} = 200 \text{ km} + 50 \text{ km} = 250 \text{ km} $$
• Total time spent so far = Time (Leg 1) + Time (Lag 2)
$$ \text{Time Spent} = 2 \text{ h} + 1 \text{ h} = 3 \text{ hours} $$
• Remaining distance:
$$ \text{Remaining Distance} = \text{Total Distance} - \text{Distance Covered} = 400 \text{ km} - 250 \text{ km} = 150 \text{ km} $$
• Remaining time:
$$ \text{Remaining Time} = \text{Total Time} - \text{Time Spent} = 8 \text{ h} - 3 \text{ h} = 5 \text{ hours} $$

Calculating Speed for Remaining Distance

Finally, we calculate the speed needed to cover the remaining distance within the remaining time.

$$ \text{Speed}_{\text{remaining}} = \frac{\text{Remaining Distance}}{\text{Remaining Time}} $$

Plugging in the calculated values:

$$ \text{Speed}_{\text{remaining}} = \frac{150 \text{ km}}{5 \text{ h}} = 30 \text{ km/h} $$

Therefore, Rashi needs to travel at a speed of 30 km/h for the remaining distance.