A person takes a trip, driving with a constant speed 85.6 km/h except for a 22.6 minute rest stop.
If the person's average speed is 61.6 km/h, how far is the trip? Answer in units of km.
Let T be the total time (in hours) and X be the distance (in km). 22.6 minutes is 0.3767 hours. You can write two equations in the two unknowns:
X = 61.6 T (using the average speed)
and
X = 85.6 (T-.03767)
(using the time spent actually driving)
This means that
61.6 T = 85.6 T - 32.24
24.0 T = 32.24
T = 1.343 h
Now solve for X
X = ?
22.6 minutes is 22.6/60 hours = 0.3737 hours
If for simplicity we take the rest stop at the start the total journey time is (0.3767 + t)
where t is the driving time.
If d is the distance travelled then speed =
d/t=85.6 km/h
However the average speed is from time zero, thus speed from time zero
is d/(t+0.3767)=61.6 km/h
Next step is to rearrange
d/t=85.6 km/h
with t as the subject and substitute in
d/(t+0.3767)=61.6 km/h
and solve for d
Hope this is OK.
To find the distance of the trip, we can use the formula:
Distance = Speed × Time
First, let's calculate the time spent driving.
Rest stop duration: 22.6 minutes
To convert this time to hours, we divide by 60 (since there are 60 minutes in an hour):
Rest stop duration in hours = 22.6 minutes / 60 minutes/hour
Rest stop duration in hours = 0.3767 hours (rounded to four decimal places)
The total time spent driving is the total time minus the rest stop duration.
Total time = Rest stop duration + Time spent driving
Total time = 0.3767 hours + (Time spent driving in hours)
Average speed = Total distance ÷ Total time
Rearranging the formula, we can solve for Total distance:
Total distance = Average speed × Total time
We know that the average speed is 61.6 km/h, and the total time is the time spent driving.
Total distance = 61.6 km/h × (Total time in hours)
To find the total time spent driving, we need to exclude the time spent at the rest stop from the total time.
Total time spent driving = Total time - Rest stop duration in hours
Now we can calculate the total distance.
Let's substitute the values into the formula:
Total distance = 61.6 km/h × (Total time - 0.3767 hours)
Since the person drives at a constant speed of 85.6 km/h, we can also say:
Total time - 0.3767 hours = Distance ÷ 85.6 km/h
Now we can write the equation using the given information:
85.6 km/h × (Total time - 0.3767 hours) = 61.6 km/h × (Total time - 0.3767 hours)
We can cancel out the hours on both sides of the equation:
85.6 km × (Total time - 0.3767) = 61.6 km × (Total time - 0.3767)
85.6 × Total time - 85.6 × 0.3767 = 61.6 × Total time - 61.6 × 0.3767
Now let's rearrange the equation to solve for Total time:
24 × Total time = 24 × 0.3767
Total time = 0.3767 hours
Finally, substitute the value of Total time into the formula to find the total distance:
Total distance = 61.6 km/h × (Total time - 0.3767 hours)
Total distance = 61.6 km/h × (0.3767 hours - 0.3767 hours)
Total distance = 61.6 km/h × 0
Therefore, the total distance of the trip is 0 km.
To find the distance of the trip, we can use the formula:
Distance = Average speed × Time
Let's break down the problem:
1. The person drives at a constant speed of 85.6 km/h.
2. During the trip, there is a rest stop of 22.6 minutes.
3. The average speed of the person throughout the entire trip is given as 61.6 km/h.
To get the distance, we need to find the total time without the rest stop, as the rest stop does not contribute to the distance traveled.
Step 1: Convert the rest stop time to hours.
Since the rest stop duration is given in minutes (22.6 minutes), we need to convert it to hours. There are 60 minutes in an hour.
So, 22.6 minutes = 22.6 / 60 = 0.3767 hours (approximately).
Step 2: Calculate the time spent driving without the rest stop.
The total time spent driving without the rest stop can be calculated using the formula:
Time driving = Total time - Rest stop time
Given that the average speed is 61.6 km/h and the total time for the trip is unknown, we can write the equation as:
Total time - 0.3767 hours = Distance / 61.6 km/h
Step 3: Calculate the total time for the trip.
Rearranging the equation from Step 2, we get:
Total time = Distance / 61.6 km/h + 0.3767 hours
Since we know that average speed is distance divided by time, we can write:
61.6 km/h = Distance / Total time
Rearranging this equation, we find:
Distance = 61.6 km/h × Total time
Step 4: Combine the equations.
Now we have two equations involving the total time for the trip:
Total time = Distance / 61.6 km/h + 0.3767 hours
Distance = 61.6 km/h × Total time
Step 5: Solve for the distance.
Substitute the first equation into the second equation:
Distance = 61.6 km/h × (Distance / 61.6 km/h + 0.3767 hours)
Simplifying the equation, we have:
Distance = Distance + (61.6 km/h × 0.3767 hours)
Subtracting Distance from both sides, we get:
0 = (61.6 km/h × 0.3767 hours)
This equation shows that Distance cancels out on both sides, which means the distance does not affect the equation.
Step 6: Calculate the rest stop time.
Now, we can solve for the rest stop time by rearranging the equation above:
Rest stop time = 0.3767 hours.
Step 7: Calculate the total time without rest stop.
The total time without the rest stop can be found by subtracting the rest stop time from the total trip time:
Total time without rest stop = Total time - Rest stop time.
Substituting the known values, we find:
Total time without rest stop = Total time - 0.3767 hours.
Step 8: Solve for the total time.
Since we have another equation involving the total time, let's substitute the expression for total time without rest stop into the equation:
Total time = Distance / 61.6 km/h + 0.3767 hours.
Substituting the expression for total time without rest stop, we get:
Total time without rest stop = (Distance / 61.6 km/h + 0.3767 hours) - 0.3767 hours.
Simplifying the equation, we have:
Total time without rest stop = Distance / 61.6 km/h.
Step 9: Calculate the distance.
With the two equations:
Total time without rest stop = Distance / 61.6 km/h.
and
Total time without rest stop = Total time - Rest stop time, which we found to be 0.3767 hours.
We can substitute the value for Total time without rest stop:
Distance / 61.6 km/h = Total time - Rest stop time.
Substituting the known value, we find:
Distance / 61.6 km/h = Total time - 0.3767 hours.
Finally, substituting the given average speed of 61.6 km/h, we have:
Distance / 61.6 km/h = 61.6 km/h - 0.3767 hours.
Simplifying further, we find:
Distance = (61.6 km/h - 0.3767 hours) × 61.6 km/h.
Calculating this expression will give you the distance of the trip.