i usually catch the bus home from work.i get on at the first stop and have about 10 km to travel ,so i usually nod off to sleep.i normally fall asleep when the bus has about 3 times as far to go as it has already travelled.halfway through the trip i tend to wake up and then i finally fall to sleep again when the bus has 3 quarters of the remaining distance to go.i wake up just in time to get off at the right bus stop.what portion of the trip do i usually sleep?

Question

i usually catch the bus home from work.i get on at the first stop and have about 10 km to travel ,so i usually nod off to sleep.i normally fall asleep when the bus has about 3 times as far to go as it has already travelled.halfway through the trip i tend to wake up and then i finally fall to sleep again when the bus has 3 quarters of the remaining distance to go.i wake up just in time to get off at the right bus stop.what portion of the trip do i usually sleep?

Answer 1

To determine what portion of the trip you usually sleep, we need to first understand the total distance of the trip and the specific points at which you fall asleep and wake up.

Let's break down the information provided:

- Total distance: 10 km
- You fall asleep when the bus has about 3 times as far to go as it has already traveled.
- You wake up halfway through the trip.
- You fall asleep again when the bus has 3/4 (or 75%) of the remaining distance to go.
- You wake up just in time to get off at the right bus stop.

Step 1: Calculate the halfway point of the trip.
Halfway through the trip would be at a distance of 10 km / 2 = 5 km.

Step 2: Calculate the distance you sleep initially.
You fall asleep when the bus has about 3 times as far to go as it has already traveled, which means you fall asleep when the bus has traveled 5 km / 3 = 1.67 km (approximately).

Step 3: Calculate the remaining distance at your first waking point.
At the halfway point (5 km), you wake up. So, the remaining distance at this point would be 10 km - 5 km = 5 km.

Step 4: Calculate the distance you sleep during the second phase.
You fall asleep again when the bus has 3/4 of the remaining distance to go, which is 3/4 * 5 km = 3.75 km (approximately).

Step 5: Calculate the distance you are awake during the second phase.
At this point, there is a remaining distance of 5 km - 3.75 km = 1.25 km.

Step 6: Calculate the total distance you are asleep.
The total distance you sleep is the distance you sleep initially (1.67 km) plus the distance you sleep during the second phase (3.75 km), which equals 1.67 km + 3.75 km = 5.42 km.

Step 7: Calculate the portion of the trip you usually sleep.
To find the portion of the trip you usually sleep, divide the total distance you sleep (5.42 km) by the total distance of the trip (10 km) and multiply by 100%:

(5.42 km / 10 km) * 100% = 54.2%

Therefore, you usually sleep approximately 54.2% of the trip.