A wheel 1 meter in diameter is rotating at 100 revolutions per minute without slipping along a level stretch of road and passes mile marker 0 at 12:00 noon. it continues past mile marker 0 for 10 minutes and then has a period of angular acceleration of 2.5x10^-2 radians per second^2 for a period of 90 seconds. After the 90 sec acceleration period, the wheel rolls on for 1 hour. At the end of the hour the wheel undergoes a deceleration of 1.5x10^-2 radians per sec^2 until it stops. How far is the wheel in km from mile marker 0? what time is it when it stops?

To solve this problem, we need to break it down into smaller steps:

1. Find the distance covered during the initial 10-minute period.
2. Determine the final angular velocity after the 90-second acceleration.
3. Calculate the distance covered during the 1-hour period.
4. Determine the time it takes for the wheel to come to a stop.
5. Add up all the distances to find the total distance from mile marker 0.
6. Calculate the time when the wheel stops.

Let's begin with step 1:

1. Distance covered during the initial 10-minute period:
Given that the wheel is rotating at 100 revolutions per minute and its diameter is 1 meter, we can calculate the circumference of the wheel using the formula C = πd, where C is the circumference and d is the diameter.
The circumference is C = π(1) = π meters.
In 10 minutes, the wheel will make 10/60 = 1/6 of a revolution.
The distance covered during this time is (1/6) * C = (1/6) * π meters.
To convert this distance to kilometers, divide by 1000: (1/6) * π / 1000 = π/6000 kilometers.

Next, let's move on to step 2:

2. Final angular velocity after the 90-second acceleration:
We know the initial angular velocity is 100 revolutions per minute, which can be converted to radians per second by multiplying by 2π/60.
The initial angular velocity is (100 * 2π/60) radians per second.
The angular acceleration is 2.5x10^-2 radians per second^2.
Using the formula: Final angular velocity = Initial angular velocity + (angular acceleration * time),
we can find the final angular velocity after 90 seconds:
Final angular velocity = (100 * 2π/60) + (2.5x10^-2 * 90) radians per second.

Now, let's proceed to step 3:

3. Distance covered during the 1-hour period:
Given that the wheel is now rolling without any angular acceleration, its angular velocity remains constant.
The distance covered by the wheel in 1 hour is the product of the angular velocity (found in step 2) and the time in seconds:
Distance = angular velocity * time = Final angular velocity * (1 hour * 60 minutes * 60 seconds).

Moving on to step 4:

4. Time it takes for the wheel to come to a stop:
The wheel undergoes deceleration of 1.5x10^-2 radians per second^2 until it stops.
We need to find how long it takes for the wheel to come to a stop.
The formula to calculate the time is: Final angular velocity = 0 = Initial angular velocity + (angular acceleration * time).

Now, let's proceed to step 5:

5. Total distance from mile marker 0:
Add up the distances covered in step 1, step 3, and any additional distances covered during the angular deceleration period.

Finally, let's calculate step 6:

6. Time when the wheel stops:
Add the time taken in step 1, the time taken in step 3, and the time taken to come to a stop (from step 4) to the starting time of 12:00 noon.

Once all these calculations are performed, we will have the answers to both questions: the distance of the wheel from mile marker 0 in kilometers and the time when it stops.