Two courier vans leave depots A and B at the same moment and drive toward one another. The depots are 300 km apart. The van from A travels at 30 km/h, while the van from B travels at 46 km/h. After the vans meet, how long will the van from B take to reach depot A? Give your answer to two decimal places.

The correct option is 2.57 hours.

To find how long the van from B takes to reach depot A after the two vans meet, we can break down the problem into three simple steps:
1. Calculate the time it takes for both vans to meet.
2. Determine the remaining distance van B needs to travel to reach depot A.
3. Calculate the time van B takes to cover that remaining distance.

Step 1: Find the time taken for the vans to meet
The total distance between depot A and depot B is 300 km.
Since both vans drive toward each other, their combined (relative) speed is the sum of their individual speeds:

$\text{Relative speed}=30\text{km/h}+46\text{km/h}=76\text{km/h}$

The time ($t$) taken for the two vans to meet is:

$t=\frac{\text{Total Distance}}{\text{Relative Speed}}=\frac{300}{76}=\frac{75}{19}\text{hours}$

Step 2: Find the distance remaining for van B to reach depot A
When the vans meet, the distance remaining for van B to reach depot A is exactly equal to the distance that van A has traveled up to the meeting point.
Distance traveled by van A in $\frac{75}{19}$ hours:

${\text{Distance}}_{\text{A}}=\text{Speed of A}\times t=30\times \frac{75}{19}=\frac{2250}{19}\text{km}$

Step 3: Calculate the time taken by van B to complete its journey to depot A
Van B must cover this remaining distance of $\frac{2250}{19}\text{km}$ at its speed of 46 km/h:

$\text{Time required}=\frac{{\text{Distance}}_{\text{A}}}{\text{Speed of B}}=\frac{\frac{2250}{19}}{46}=\frac{2250}{19\times 46}=\frac{2250}{874}$

Dividing the values:

$\text{Time required}\approx 2.57437\text{hours}$

Rounding to two decimal places, the van from B takes 2.57 hours to reach depot A after meeting.