Al and Bill live at opposite ends of the same street. Al had to deliver a parcel to Bill’s home, Bill one at Al’s home. They started at the same moment, each walked at constant speed and returned home immediately after leaving the parcel at its destination. They met the first time at a distance of a yards from Al’s home and the second time at a distance of b yards from Bill’s home.
Assume that Al and Bill walk less than twice as fast as each other.
I answered the question above but the second part is this and I don't know how to solve this:
How should we change the four equations we use to describe the prob- lem if Al walks more than twice as fast as Bill?
Twice as fast mean Al will be back home (there and back) when Bill reaches Al's end (ie, they both get there at the same time. That is the second time they meet, the first was at 2/3 the way from Al to Bill's end.
If Al walks more than twice as fast as Bill, we can change the four equations we use to describe the problem as follows:
Let:
- a = distance from Al's home where they first meet
- b = distance from Bill's home where they second meet
- r = Al's walking speed (rate)
- s = Bill's walking speed (rate)
- d = total distance of the street
1) From Al's home to the meeting point (first meeting):
Distance traveled by Al = a
Distance traveled by Bill = d - a (remaining distance from the meeting point to Bill's home)
Time taken by Al = a / r
Time taken by Bill = (d - a) / s
2) From Bill's home to the meeting point (second meeting):
Distance traveled by Al = d - b (remaining distance from the meeting point to Al's home)
Distance traveled by Bill = b
Time taken by Al = (d - b) / r
Time taken by Bill = b / s
Please note that the equations are still subjected to the constraint that Al and Bill walk at constant speeds and return home immediately after leaving the parcel at its destination.
To solve this problem, we need to define some variables and equations based on the given information. Let's assume that Al's walking speed is represented by "x" and Bill's walking speed is represented by "y".
Now, let's consider the scenario where Al walks more than twice as fast as Bill. This means that Al's speed is greater than 2 times Bill's speed, so we can write the inequality:
x > 2y
Based on this inequality, we need to modify the four equations we use to describe the problem.
1. Equation representing the distance covered by Al when they meet for the first time:
Distance covered by Al = a
Since Al walks at speed x for some time t, we can write: Distance covered by Al = xt
2. Equation representing the distance covered by Bill when they meet for the first time:
Distance covered by Bill = a
Since Bill walks at speed y for the same time t, we can write: Distance covered by Bill = yt
3. Equation representing the distance covered by Al when they meet for the second time:
Distance covered by Al = b
Since Al walks at speed x for some time t', we can write: Distance covered by Al = xt'
4. Equation representing the distance covered by Bill when they meet for the second time:
Distance covered by Bill = b
Since Bill walks at speed y for the same time t', we can write: Distance covered by Bill = yt'
So, when Al walks more than twice as fast as Bill, the four equations for this scenario are:
1. xt = a (Distance covered by Al when they meet for the first time)
2. yt = a (Distance covered by Bill when they meet for the first time)
3. xt' = b (Distance covered by Al when they meet for the second time)
4. yt' = b (Distance covered by Bill when they meet for the second time)
These equations can be used to solve the problem under the given conditions.