The passenger bus starts out and travels at 65 kilometers per hour. three hours later, a car starts out from the same place and overtakes the bus in 8 hours and 40 minutes. what is the rate of the car?

1 h = 60 min

40 min = 40 / 60 = 2 * 20 / ( 3 * 20 ) = 2 / 3 h

8 h 40 min = 8 h + 2 / 3 h = 8 * 3 / 3 + 2 / 3 = 24 / 3 - 2 / 3 = 26 / 3 h

3 h = 3 * 3 / 3 = 9 / 3 h

Car travel 8 h 40 min.

For this time the bus travel = 8 h 40 min + 3 h = 11 h 40 min = 8 h 40 min + 3 h = 26 / 3 h + 3 h = 26 / 3 h + 9 / 3 h = 35 / 3 h

The bus travel distance = 65 km / h * 11 h 40 min = 65 km / h * 35 / 3 h = 65 * 35 / 3 = 2275 / 3 = ( 2274 + 1 ) / 3 = 2274 / 3 + 1 / 3 = 758 + 1 / 3 km

Velocity of car = 758 km + 1 / 3 km * 8 h 40 min = ( 2275 / 3 km ) / ( 26 / 3 h ) = ( 2275 * 3 km ) / ( 26 / 3 h ) = 2275 / 26 = 87.5 km / h

Proof :

The bus travel 758 + 1 / 3 km

Car travel : 87.5 km / h * 8 h 40 min = 87.5 km / h * 8 h 40 min = ( 87.5 * 2 / 2 ) * ( 26 / 3 ) = ( 175 / 2 ) * ( 26 / 3 ) = 175 * 26 / ( 2 * 3 ) = 4550 / 6 = ( 4548 + 2 ) / 6 = 4548 / 6 + 2 / 6 = 758 + 2 / 6 = 758 + 2 / ( 2 * 3 ) = 758 + 1 / 3 km

To solve this problem, we can use the formula:

Distance = Rate × Time

Let's break down the information given:

Passenger bus:
- Rate = 65 kilometers per hour

Car:
- Time = 8 hours and 40 minutes, which is equivalent to 8.67 hours (40/60 = 0.67)
- The car starts three hours after the bus

We need to find the rate of the car.

Now, let's calculate the distance covered by the bus in the 3 hours before the car starts:
Distance(bus) = Rate(bus) × Time(bus)
Distance(bus) = 65 km/h × 3 hours
Distance(bus) = 195 km

When the car starts, it needs to cover this additional distance to catch up with the bus.

Let's denote the rate of the car as Rate(car).
The time for the car to catch up with the bus is Time(car) = 8.67 hours.

Now, let's calculate the total distance covered by the car:
Distance(car) = Distance(bus) + Distance(catch-up)

Since the car catches up with the bus, the distances covered by both vehicles are equal.
Distance(car) = Distance(bus)
Rate(car) × Time(car) = 195 km

Substituting the value of Time(car) and solving for Rate(car):
Rate(car) × 8.67 hours = 195 km
Rate(car) = 195 km / 8.67 hours
Rate(car) ≈ 22.48 kilometers per hour

Therefore, the rate of the car is approximately 22.48 kilometers per hour.