A tourist has planned a trip to cover the distance of 640 miles, driving at some constant speed. However, when he already covered a quarter of the distance, he took a rest for 1.2 hours. Then, in order to arrive at the destination on time, he increased the speed by 20 mph. How long, actually, the trip lasted?
A quarter of the distance = 640 / 4 = 160 miles
rest of the distance = 640 - 160 = 480 miles
Distance, speed, time formula:
s = d / t
If he did not make a rest he would travel:
t = 640 / s1 [ hours ]
where s1 = speed in a first quarter of the distance
t1 = d / s1 = 160 / s1
where t1 =time in a first quarter of the distance
time of a rest = 1.2 [ hours ]
New speed:
s2 = s1 + 20
New time = t2 = rest of the distance / new speed
t2 = 480 / ( s1 + 20 )
In order to arrive at the destination on time mean a trip will last:
640 / s1
how long it would take to travel at a constant speed to lasted s1
Total time:
t = 160 / s1 + 1.2 + 480 / ( s1 + 20 ) = 640 / s1
[ 160 ∙ ( s1 + 20 ) + 1.2 ∙ s1 ∙ ( s1 + 20 ) + 480 ∙ s1 ] / [ s1 ∙ ( s1 + 20 ) ] = 640 / s1
( 160 s1 + 3200 + 1.2 s1² + 24 s1+ 480 s1 ) / [ s1 ∙ ( s1 + 20 ) ] = 640 ∙ ( s1 + 20 ) / [ s1 ∙ ( s1 + 20 ) ]
( 1.2 s1² + 664 s1+ 3200 ) / [ s1 ∙ ( s1 + 20 ) ] = 640 ∙ ( s1 + 20 ) / [ s1 ∙ ( s1 + 20 ) ]
Multiply both sides by s1 ∙ ( s1 + 20 )
1.2 s1² + 664 s1 + 3200 = 640 ∙ ( s1 + 20 )
1.2 s1² + 664 s1 + 3200 = 640 s1 + 12800
Subtract 640 s1 + 12800 to both sides
1.2 s1² + 664 s1 + 3200 - 640 s1 - 12800 = 0
1.2 s1² + 24 s1 - 9600 = 0
Divide both sides by 1.2
s1² + 20 s1 - 8000 = 0
The solutions are:
s1 = - 100
and
s1 = 80
Speed can't be negative so:
s1 = 80 miles / hour
Total time:
t = 160 / s1 + 1.2 + 480 / ( s1 + 20 )
t = 160 / 80 + 1.2 + 480 / ( 80 + 20 )
t = 2 + 1.2 + 480 / 100
t = 2 + 1.2 + 4.8
t = 8 hours
To find out how long the trip actually lasted, we can break down the problem into steps.
Step 1: Determine the distance traveled before taking a rest.
Since the tourist took a rest after covering a quarter of the distance, we can calculate that as 640 miles * 1/4 = 160 miles.
Step 2: Calculate the time spent taking a rest.
We are given that the tourist rested for 1.2 hours.
Step 3: Calculate the remaining distance.
The distance remaining after taking a rest is the total distance minus the distance traveled before the rest.
Remaining distance = 640 miles - 160 miles = 480 miles.
Step 4: Calculate the time taken to cover the remaining distance at an increased speed.
Since the tourist increased the speed by 20 mph, we need to calculate the time taken to cover the remaining distance of 480 miles at the increased speed.
Time = Distance / Speed
Time = 480 miles / (original speed + 20 mph)
Step 5: Calculate the total time for the trip.
The total time for the trip is the time spent before the rest, the time spent resting, and the time taken to cover the remaining distance at an increased speed.
Total time = Time spent before the rest + Time spent taking a rest + Time taken to cover the remaining distance at an increased speed
Now, let's perform the calculations:
Step 1: Distance traveled before taking a rest: 160 miles
Step 2: Time spent taking a rest: 1.2 hours
Step 3: Remaining distance: 480 miles
Step 4: Time taken to cover the remaining distance at an increased speed: 480 miles / (original speed + 20 mph)
Step 5: Total time for the trip: (160 miles / original speed) + 1.2 hours + (480 miles / (original speed + 20 mph))
Unfortunately, we don't have the original speed provided, so we cannot provide the exact total time for the trip. However, if you provide the original speed, we can calculate the total time using the formula provided in step 5.
To find out how long the trip actually lasted, we need to calculate the time it took for each portion of the trip and then add them together.
Let's break down the trip into three parts:
1. The initial quarter of the trip: The tourist covered 640 miles / 4 = 160 miles. Since we don't know the speed during this part, let's call it "x" mph. Time taken for this part can be calculated using the formula: time = distance / speed. So, the time taken would be 160 miles / x mph.
2. The second part after the rest: The tourist increased the speed by 20 mph, so the speed during this part would be (x + 20) mph. The distance remaining to be covered is 640 - 160 = 480 miles. Time taken for this part can be calculated using the same formula: time = distance / speed. So, the time taken would be 480 miles / (x + 20) mph.
3. Resting time: The tourist took a rest for 1.2 hours.
Now, to find the total time taken for the entire trip, we need to add the times of each part:
Total time = time taken for the initial quarter + time taken for the second part + resting time
Total time = (160 miles / x mph) + (480 miles / (x + 20) mph) + 1.2 hours
This equation will give us the total time taken for the trip. However, without knowing the value of x (the initial speed), we cannot find the exact time. If you have any additional information, such as the average speed or the time taken for the entire trip, we can solve for x and find the total time.