Tom and Joel are good runners, both able to run at a constant speed of 10 mph. Their amazing dog Trot can do even better, he runs at 20 mph. Starting from towns 60 miles apart, tom and joel run toward each other while Trot runs back and forth between them. How far does Trot run by the time the boys meet? Assume that Trot started with Tom running toward Joel and that he is able to make instants turnarounds.
Solve it using a geometric series then solve it an easier way....
I'm never good at these kinds of questions.
How do i do this?
Solve the problem two ways.
(a) Use a geometric series.
(b) Find a shorter way to do the problem.
Based on dog cycles, the time between dog turn arounds.
If the distance between the runners is d,
(2/3) d (1/3) d
*-----------------------------|--------------*
(a) The dog will run 2/3 of the distance
(b) Each runner will run 1/3 of the distance.
(c) The new distance will be d minus the distance the runners ran.
(d) The dog will run (2/3) d.
(e) The new distance will be (1/3) d.
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Distance d d/3 d/3^2 d/3^3 ....
The dog will run
(2/3)d + (2/3)(d/3) + (2/3) (d/3^2) + (2/3) (d/3^3) + ......
(2/3) d
(2/3) d ( 1 + 1/3 + 1/3^2 + 1/3^3 + ,.....) = -----------
1-1/3
2d
= ----------- = d = 60 miles.
3-1
(b) The runners have to run for 3 hours. The dog runs for
3 hours so he runs 60 miles.
or The dog runs twice as fast as the runner. Each runner runs 30
miles, so the dog runs sixty miles.
The geometric series method is straightforward and you need to practice setting up such methods yourself....
To solve it the easy way, suppose there was another dog Trot2 who started out running with Trot, however Trot2 forgets to make the turns and just continues to run in a straight line. How far will Trot2 have run by the time the two runners meet. And at any given time what is the relation between the total distance run by Trot and Trot2?
To solve this problem using a geometric series, you can consider the distances covered by each runner in terms of time. Let's break it down step by step:
Step 1: Determine the time it takes for Tom and Joel to meet.
Since they are running towards each other, their combined speed is 10 mph + 10 mph = 20 mph. Therefore, the time it takes for Tom and Joel to meet is given by:
Time = Distance / Speed
Time = 60 miles / 20 mph
Time = 3 hours
Step 2: Calculate the total distance Trot runs during this time.
Since the boys take 3 hours to meet, Trot will also take 3 hours to run back and forth between them. However, to calculate the total distance Trot runs, we need to know how many round trips he makes in these 3 hours.
Step 3: Determine the number of round trips Trot makes.
During the time it takes for Tom and Joel to meet, Trot's speed of 20 mph allows him to cover twice the distance the boys cover. Therefore, Trot's round trip distance is 2 times the distance covered by the boys during the meeting time.
Round Trip Distance = 2 * Distance covered by Tom and Joel
Round Trip Distance = 2 * (Speed * Time)
Round Trip Distance = 2 * (10 mph * 3 hours)
Round Trip Distance = 2 * 30 miles
Round Trip Distance = 60 miles
Step 4: Calculate the total distance Trot runs.
Since Trot's round trip distance is 60 miles and he takes 3 hours to complete each round trip, the total distance Trot runs is given by:
Total Distance = Round Trip Distance * Number of Round Trips
Total Distance = 60 miles * Number of Round Trips
Now, to find the number of round trips Trot makes, we use the fact that he runs for the same amount of time it takes for Tom and Joel to meet:
Number of Round Trips = Time / Round Trip Time
Number of Round Trips = 3 hours / 3 hours
Number of Round Trips = 1
Therefore, the total distance Trot runs is:
Total Distance = 60 miles * 1
Total Distance = 60 miles
So, Trot runs a total of 60 miles by the time the boys meet.
Now, let's solve the problem in an easier way:
In order to find how far Trot runs, we need to determine the total time it takes for the boys to meet. We have already established that the time is 3 hours.
Since Trot runs back and forth the same distance while Tom and Joel are running towards each other, we can simply calculate the distance Trot runs in one direction and double it.
Trot's speed is 20 mph and the time taken for the boys to meet is 3 hours. Hence, the distance Trot runs in one direction is given by:
Distance = Speed * Time
Distance = 20 mph * 3 hours
Distance = 60 miles
Since Trot makes round trips and the distance in one direction is 60 miles, the total distance Trot runs will be:
Total Distance = Distance * Number of Round Trips
Total Distance = 60 miles * 1 round trip
Total Distance = 60 miles
Therefore, regardless of the method used to solve the problem, Trot runs a total of 60 miles by the time the boys meet.