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?
why not post your solution to the first part, and we can see why this change is important?
To solve this problem, let's start by defining some variables:
Let's say the total distance between their homes is D.
Let the speed of Al be a and the speed of Bill be b.
From the information given in the problem, we can deduce the following:
1. The time taken by Al to reach the first meeting point is the same as the time taken by Bill to reach the second meeting point. Thus, the ratio of distances traveled by Al and Bill in their first trips is the same as the ratio of their speeds. This can be represented by the equation: a*(D - a) / b*(b - D) = a / b.
2. The time taken by Bill to reach the first meeting point is the same as the time taken by Al to reach the second meeting point. Thus, the ratio of distances traveled by Bill and Al in their first trips is the same as the ratio of their speeds. This can be represented by the equation: b*(b - D) / a*(D - a) = b / a.
3. The total distance they have traveled in their first trips is equal to D. This can be represented by the equation: a*(D - a) + b*(b - D) = D.
4. The total distance they have traveled in their second trips is also equal to D. This can be represented by the equation: b*(D - b) + a*(a - D) = D.
Now, let's address the second part of the question: "How should we change the four equations we use to describe the problem if Al walks more than twice as fast as Bill?"
If Al walks more than twice as fast as Bill, it means that the ratio of their speeds is greater than 2:1. In this case, Equation 1 and Equation 2 can remain the same, as they represent the equality of time taken by Al and Bill to reach the meeting points.
However, Equation 3 and Equation 4 need to be modified. Since Al is now walking more than twice as fast as Bill, the distance traveled by Al in his first trip will be greater than the distance traveled by Bill. Thus, Equation 3 should be changed to: a*(D - a) + b*(b - D) < D.
Similarly, in Equation 4, the distance traveled by Al in his second trip will be less than the distance traveled by Bill. So Equation 4 should be changed to: b*(D - b) + a*(a - D) < D.
These modified equations reflect the changed scenario where Al walks more than twice as fast as Bill.