Emily takes a trip, driving with a constant velocity of 79.5 km/h to the north except for a 22 min rest stop. If Emily's average velocity is 68.8 km/h to the north, how long does the trip take?
To find the total time for the trip, we first need to find the distance traveled.
Since Emily's average velocity is 68.8 km/h to the north, we can use this value to find the distance traveled during the trip.
We can use the formula:
Distance = Velocity * Time
Let's assume the total time for the trip is t hours.
So, the distance traveled during the trip can be calculated as:
Distance = 68.8 km/h * t
Now, let's consider the rest stop. Emily takes a 22-minute rest stop, which is equal to (22/60) hours.
Given that Emily's velocity during the trip (excluding the rest stop) is 79.5 km/h, we can calculate the distance traveled during this time as:
Distance = 79.5 km/h * (t - 22/60) hours
Since the total distance traveled is the same, we can equate the two equations:
68.8 km/h * t = 79.5 km/h * (t - 22/60)
Let's solve this equation step-by-step to find the value of t.
Step 1: Distribute the terms on the right side of the equation
68.8 km/h * t = 79.5 km/h * t - (79.5 km/h * 22/60)
Step 2: Collect like terms
68.8 km/h * t - 79.5 km/h * t = - (79.5 km/h * 22/60)
(-10.7 km/h) * t = - (79.5 km/h * 22/60)
Step 3: Divide both sides of the equation by -10.7 km/h to isolate t
t = - (79.5 km/h * 22/60) / (-10.7 km/h)
t = (79.5 * 22) / (60 * 10.7)
Now, let's calculate the value of t.
t ≈ 26.69 hours
Therefore, the trip takes approximately 26.69 hours.
To find the total time of the trip, we can break it down into two parts: the time spent driving and the time spent at the rest stop.
Let's start by calculating the time spent driving. We'll use the formula:
Time = Distance / Velocity
Given that Emily drives with a constant velocity of 79.5 km/h, we need to determine the distance traveled. However, we don't have the distance directly given. Instead, we will use the average velocity to calculate it.
We know that average velocity is defined as the total distance traveled divided by the total time taken. So, we can rearrange this equation to solve for the distance:
Distance = Average Velocity * Total Time
Now, we can substitute the values into the equation:
Distance = 68.8 km/h * Total Time
Next, we'll consider the time spent at the rest stop, which is given as 22 minutes.
Now, let's put everything together. The total time of the trip can be calculated as:
Total Time = Time Driving + Time at Rest Stop
Total Time = Distance / Velocity + Rest Stop Time
Substituting the calculated values:
Total Time = (68.8 km/h * Total Time) / 79.5 km/h + 22 min
To solve this equation, we can isolate the Total Time term:
Total Time - (68.8 km/h * Total Time) / 79.5 km/h = 22 min
Multiplying through by 79.5 km/h:
79.5 km/h * Total Time - 68.8 km/h * Total Time = 22 min * 79.5 km/h
10.7 km/h * Total Time = 22 min * 79.5 km/h
Now, we need to convert the units. 22 minutes can be converted to hours by dividing by 60:
10.7 km/h * Total Time = (22/60) h * 79.5 km/h
10.7 km/h * Total Time = 8.25 km
Dividing both sides of the equation by 10.7 km/h:
Total Time = 8.25 km / 10.7 km/h
Simplifying:
Total Time = 0.77 h
Therefore, the trip takes approximately 0.77 hours, or 46.2 minutes.