Stan takes a trip, driving with a constant velocity of 81.5 km/h to the north except for a 22 min rest stop. If Stan's avg velocity is 66.80 km/h to the north, how long does the trip take?
Let the driving time be t hours.
The distance covered is 81.5t km.
However, if we factor in the rest stop of 22/60 hours, the distance is 66.8(t+22/60).
Equating the two distances, we get
66.8(t+22/60) = 81.5t
66.8t + 25 = 81.5t
25 = 14.7t
t = 1.7 hours driving, so
the trip took 1hr 42min + 22min = 2hr 4min
To find the total time of the trip, we need to consider both the time spent driving and the time spent during the rest stop.
We are given that Stan's average velocity during the whole trip is 66.80 km/h. This is the total distance traveled divided by the total time taken. Let's call the total distance traveled as "d" and the total time taken as "t".
The average velocity is given by:
Average Velocity = Total Distance / Total Time
We know that the average velocity during the driving part of the trip is 81.5 km/h. We can use this information to find the distance traveled during the driving part of the trip.
Distance Traveled = Average Velocity * Time Spent Driving
We are also given that there was a 22-minute rest stop during the trip. This means that the total time taken is the sum of the time spent driving and the time spent during the rest stop.
Total Time = Time Spent Driving + Time Spent Resting
We can now set up two equations based on the information given:
Equation 1: 66.80 km/h = d / t (using the average velocity equation)
Equation 2: d = 81.5 km/h * Time Spent Driving (using the distance traveled equation)
To find the time of the trip, we need to solve these equations simultaneously.
Substituting Equation 2 into Equation 1, we get:
66.80 km/h = (81.5 km/h * Time Spent Driving) / t
Simplifying, we have:
66.80 km/h * t = 81.5 km/h * Time Spent Driving
Dividing both sides of the equation by 66.80 km/h, we get:
t = (81.5 km/h * Time Spent Driving) / 66.80 km/h
Now, let's consider the time spent resting. Since we know that the rest stop took 22 minutes, we can subtract this time from the total time to get the time spent driving.
Time Spent Driving = Total Time - Time Spent Resting
Time Spent Driving = t - 22 minutes
Substituting this into the previous equation, we get:
t = (81.5 km/h * (t - 22 minutes)) / 66.80 km/h
Now we can solve the equation to find t.
First, let's convert the time in minutes to hours by dividing by 60:
22 minutes = 22/60 = 0.37 hours
t = (81.5 km/h * (t - 0.37 hours)) / 66.80 km/h
To solve this equation, we can cross-multiply:
t * 66.80 km/h = 81.5 km/h * (t - 0.37 hours)
Expanding the equation:
66.80 t = 81.5 t - 81.5 * 0.37 hours
Rearranging and solving for t:
66.80 t - 81.5 t = -81.5 * 0.37 hours
-14.7 t = -30.155 hours
Dividing both sides by -14.7:
t = -30.155 / -14.7
t ≈ 2.048 hours
Therefore, the trip takes approximately 2.048 hours.