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 it takes for Emily's trip, we will first determine the time it takes for her to travel without the rest stop.
Let's assume the duration of the trip without the rest stop is t hours.
The distance covered by Emily during this duration can be calculated using the formula:
Distance = Speed * Time
Given that her velocity is 79.5 km/h and she travels for t hours, the distance she covers is 79.5t km.
Next, we need to consider the rest stop of 22 minutes. Since 1 hour is equivalent to 60 minutes, the rest stop duration in hours can be calculated as:
Rest Stop Duration = 22 minutes / 60
Rest Stop Duration = 0.367 hours
Since her average velocity for the entire trip is 68.8 km/h, we can calculate the total distance covered with the rest stop as:
Total Distance = Average Velocity * Total Time
Total Distance = 68.8 km/h * (t + 0.367)
Since the distance covered during the trip remains the same (79.5t km), we can write the equation:
79.5t = 68.8 * (t + 0.367)
Simplifying the equation, we get:
79.5t = 68.8t + 25.168
11.7t = 25.168
t = 2.153 hours
Therefore, the time it takes for Emily's trip, including the rest stop, is approximately 2.153 hours, or 2 hours and 9.18 minutes.
To find the total time for the trip, you need to consider both the time spent driving and the time spent at the rest stop.
Let's first calculate the time spent driving. We know that Emily's average velocity during the entire trip is 68.8 km/h to the north, and we also know that her constant velocity is 79.5 km/h to the north. Since the constant velocity is faster than the average velocity, it means the rest stop time is affecting the average velocity.
To calculate the time spent driving, we can use the formula:
Time = Distance / Velocity
The distance travelled during the driving portion of the trip remains the same, regardless of the rest stop. So let's assume the distance is D and the time spent driving is T.
T = D / 79.5 km/h
Now, let's consider the impact of the rest stop on the average velocity. We know that during the driving portion of the trip, Emily's velocity was 79.5 km/h to the north. But during the rest stop, her velocity is zero since she is not moving. The rest stop duration is 22 minutes, which is equal to 22/60 = 0.367 hours.
To find the distance travelled during the driving portion, we can use the formula:
Distance = Average Velocity x Total Time
Since the average velocity is 68.8 km/h and the total time is T, we have:
D = 68.8 km/h x T
Now, let's consider the rest stop duration's effect on the average velocity. When Emily is at rest, her average velocity is zero. Therefore, we can write the equation for overall average velocity as:
68.8 km/h = (D / (T + 0.367 hours))
By replacing the value of D, the equation becomes:
68.8 km/h = ((68.8 km/h x T) / (T + 0.367 hours))
To solve this equation, we can rearrange it to isolate T:
(68.8 km/h) x (T + 0.367 hours) = (68.8 km/h x T)
(68.8 km/h) x T + (68.8 km/h) x 0.367 hours = (68.8 km/h x T)
(68.8 km/h) x 0.367 hours = (68.8 km/h) x T - (68.8 km/h) x T
(68.8 km/h) x 0.367 hours = 0
Since the equation on the right side results in zero, it means the left side of the equation must also equal zero:
(68.8 km/h) x 0.367 hours = 0
Now we can solve for T:
T = (68.8 km/h) x 0.367 hours / 68.8 km/h
T = 0.367 hours
Therefore, the time spent driving is 0.367 hours.
Now, to calculate the total time of the trip, we need to add the rest stop time to the driving time:
Total time = Time spent driving + Rest stop time
Total time = 0.367 hours + 0.367 hours
Total time = 0.734 hours
Since we need the answer in minutes, let's convert 0.734 hours to minutes:
Total time = 0.734 hours x 60 minutes/hour
Total time = 44.04 minutes
So, the total time for the trip is approximately 44.04 minutes.