Problem of the Week
for Friday, November 7

 

One more question about those ants:
(Recall that the ants have gotten into Mr. Swacker's candy supply and their trail runs from the door of 9-1 straight to the candy cupboard. Mr. Swacker has placed his pet anteater, Boris, outside the door and any ants that come our are immediately devoured. The entire trail is 12 feet long and the ants crawl at a rate of 1 inch per second. Again, when they reach the cupboard, they turn around, and when two ants bump into each other they both change direction.)

Now there are only two ants left. One is at the door headed for the cupboard, but we don't know anything about where the other ant is along the trail, nor what direction it is facing.
Is it possible to determine how long it will be before both ants are eaten? If so, how many second will it take?

Ben's Solution

If you have two ants that have travelled 72 inches so they would meet in the middle. On the sheet that has three ants one at 9-1 one in the middle and one at the cupboard it states if the ants crash into each other they immediately turn around and head their ways. If two ants were just about to meet as in this diagram:

this would be the next picture after they had hit:

but in my solution, they don't bounce their seperate way they could go as they please passing each other, if you don't count the minor set back or when they overlap like this as a bird's eye view:

With this, you could place your mysterious ant anywhere and the bouncing would not effect how long it would take.

Nicholas' Solution

It will take 288 seconds (4 mins 48 secs) before both ants are eaten.

B travels 12' from starting point to crash with A, then back to candy (X+X+12'-2X). Then another 12' from the candy to the door. Total travel = 24' = 288" = 288 seconds = 4min 48 seconds.

A travels X as B travels X to candy. B travels half way back to new position of A, crashes and goes back to candy (which is like B going all the way to A. This distance is 12' - X plus the X to start = 12'. B travels another 12' from candy to the door. Thats 24' all together.

 

Animated Solution