emmdubbs

hey all...if anyone can be of some input...

Story:
The past year has not been good for Bob. First, he lost his left hand in a serious xmas shopping accident, and his toaster stopped working properly. Specifically the middle element of the three doesn't work, tihs means the toaster can only toast half a piece of bread in each slot at the same time. So, obviously to taost a single piece of bread completely, bob must toast the first side, turn the slice around and toast the back. (or does he?). Of course bob could buy a new toaster, but he is cheap and mentally retarded, and comes to you for help instead.

After the introductions are made, bob explains to you that he always has three pieces of tast with his coffee every mornign. He further goes on to explain thath e wants to make the 3 pieces as qiuckly as possible and he has timed each step he takes to help you come up with a solution. It takes bob 30 seconds to tast a single side in one slot, 5 to take a piece out, 5 to put one back in, and if he flips it over, that takes 3 more seconds. (times are approximate)

Give bob a schedule that will be efficient as possible for him to get the three tasks done and keeping in mind that because ofh is physical impairment and mental retardation it would be IMPOSSIBLE to do 2 manual tasks at the same time...



well thats it...this is a big chunk of my final grade in math..so..any input would be greatly appreciated...
and for those of you who arn't familiar with the type of toaster it is...

it has 2 slots for 2 pieces of bread, with a "heater" on the left, center and right side of each slot, the center part is the piece that is broken, so only the outside of the bread gets toasted... this is kinda like that problem we did as kids witht he farmer, duck, wolf, and duck seed andhad to get from one side of the river to the other, but only taking one thing at a time...etc. etc.

if anyone can help, it'd be greatly appreciated.

mathmo

The most efficient method (I think) is to put two slices of toast in the toaster for 30 seconds, then remove ONE of the slices and replace with a new slice, and turn the other slice over. Toast for another 30 seconds so that one slice is toasted on both sides and the other is toasted on one side. Remove the completely toasted slice and replace with the other slice that is toasted on one side and turn over the slice that is still in the toaster, then toast for another 30 seconds and all your slices are done.

So the time taken is 30 + 5 + 5 + 3 + 30 + 5 + 5 + 3 + 30 = 116 seconds or 1 min 56 seconds.

emmdubbs

mathmo...thank you so much..thats the idea i was running with only i had done 2 pieces, switched the slots they were in avoiding the flipping...and when both those were done, putting a new piece in, and waiting, switching..etc....

your way was def. more effeciant... bravo

__________________
"You're not your job. You're not how much money you have in the bank. You're not the car you drive. You're not the contents of your wallet. You're not your fucking khakis. You're the all-singing, all-dancing crap of the world." - Tyler Durden

mathmo

Ah, I may have misread. I thought it was 3 seconds to turn the slice over in total. But it seems you mean that it takes 5 seconds to remove, 3 seconds to turn and then another 5 seconds to put it back in. That changes things ...

We have to replace everywhere I have put a 3 with 5+5+3 so this adds 20 seconds to our time, making it now 2 minutes 16 seconds. I wonder if there is a quicker way then with switching slots instead of turning, I'll have a think ... or perhaps someone might beat me to it?

Ok, I've had a think and it gets tricky depending how you interpret things.
The following is faster than the above and nasty ...

You may want a diagram, I suggest two vertical lines representing the two slots, marked off in 5 second intervals.

Let's call the slots 1 and 2 and the slices A, B and C. Put slice A in slot 1 for 30 seconds. and at the same time slice B is in slot 2, but for 15 seconds. After those 15 seconds, slice B is removed and slice C is put in slot 2 (10 seconds) 5 seconds later, we remove slice A then put slice B in slot 1 for 30 seconds. Now slice C only needs another 15 seconds and is then removed and replaced by slice A which is toasted for 30 seconds. 5 seconds after slice A is put in slot 2, slice B is removed from slot 1 and replaced by slice C. 15 seconds later slice A is removed from slot 2 and replaced by slice B for the 15 seconds that was missed at the beginning. 5 seconds later slice C is fully toasted, and another 10 seconds and slice B is toasted.

Now, we have used 2 minutes, which is the minimum since every slice needs 30 seconds to toast one side and 10 seconds to be replaced. Since there are 6 sides, this is 6*(30+10) = 240, but there are two slots, so this is 240/2 = 120 = 2 minutes.

Fingers crossed this is right...

Pellaz

put 2 peieces (a and b) in (10 sec), toast (30) remove both(10) insert bread 'a' into toaster slot 2, put bread C in slot 1(10) toast (30) remove bread a(now done) and c (10) move c to slot 2, and put b in slot 1(10) toast(30) remove(10) all done!
10+30+10+10+30+10+10+30+10=150 seconds, or 2 minutes, 30 seconds

__________________
" ' Big Mouth.
Remember it took three of you to kill me.
A god, a boy, and, last and least, a hero.' "

Pellaz

Mathmo-you missed a couple steps in your math-you never put the first bread in the toaster, and you left it in after you got done cooking it-your total time should of been 156 seconds.

edit-sorry-didn't see YOUR edit lol

__________________
" ' Big Mouth.
Remember it took three of you to kill me.
A god, a boy, and, last and least, a hero.' "

mathmo

Well, I wasn't sure if we counted adding and removing the slices at the beginning and end since the question wants the most efficient way of performing the tasks, i.e toasting the 3 slices. I assumed that the bread was magically already in the toaster and that it wasn't important to remove it ...

Hell, I'm a pure Mathematician, I don't deal with real world problems

mathmo

If we are counting the 20 seconds needed to add the bread and remove the toast, then my solution is 140 seconds, or 2 minutes 20.

Pellaz

At which point it becomes relevant what type of class this is. Mathmos solution is the mathematically correct one, but I wonder if it would be acceptable in other courses...I can think of a physics professor I had to who mathmos answer would be unacceptable, as a)things don't cook that way (you can't cook something for 30 minutes, wait an hour, cook for another 30 and say you cooked it for one hour straight) b)Mathmos toaster allows the removal of only one piece of bread at a time, while a conventional toaster will pop both trays up in unison, and shut off both heating elements for the period that the trays are 'up'. At that point, things such as the moisture content of the bread, ambient temp and temp of the toaster heating elements become relevant And without the magic toaster, Mathmos solution no longer works, because both pieces of bread are out of the toaster for 10 seconds when one is removed midway through cooking.

Just some thoughts, you can make the problem as complicated as you like, although I don't really think it needs to be. The world needs a devils advocate though.

Edited to fix some of the worse grammar mistakes

Edit #2 to fix the spelling, how sad lol

__________________
" ' Big Mouth.
Remember it took three of you to kill me.
A god, a boy, and, last and least, a hero.' "

JadziaDax

but you left the spelling errors

(just had to be said)