Crossing the Desert

Date: 05/22/2001 at 17:50:49
From: Jonathan Lefebvre
Subject: 1000-mile trip

I am working on an extra-credit problem for my honors geometry class. 
A truck gets one mile per gallon, and can hold 400 gallons at a time. 
How much is the minimum amount to cross a 1000-mile desert? 

I figured out 25,000 gallons if you deposit 100 gallons every 100 
miles for the first 600 miles, but I think there is a way to use less. 
I already tried leaving 400 gallons at the 600-mile mark, 400 gallons 
at the 400-mile mark, and 200 gallons at the 600-mile mark. These 
methods use more gas. The problem boils down to getting to the 
600-mile mark with a full tank of gas (400 gallons).

Date: 05/24/2001 at 14:36:36
From: Doctor Jaffee
Subject: Re: 1000 mile trip

Hi Jonathan,

I challenged my AP Calculus students to solve your problem using less 
than 25,000 gallons and one of my students was successful. The 
technique he used was the following.

Let's load the truck with 400 gallons, travel 100 miles, deposit 200 
gallons, and turn around and drive 100 miles back to the start. This 
process results in 200 gallons being consumed and 200 gallons being 
deposited at the 100-mile mark. Continue this process until you have 
deposited all the gas you will need at the 100-mile mark. Of course, 
we don't know how much that is yet, so let's work backwards.

As you mentioned, when we get to the 600-mile mark and we have 400 
gallons of gasoline, we can get to the end. To accomplish that we 
needed 700 gallons at the 500-mile mark for the following reason:

Suppose you are at the 500-mile mark and you have 700 gallons. Load 
400 gallons into the truck, drive 100 miles, deposit 200 gallons, and 
drive back. Now you have 300 gallons left. Load them all into the 
truck, drive 100 miles and now you have exactly 400 gallons with 400 
miles to go.

See if you can work backward, 100 miles at a time, and figure out how 
many gallons you need at the beginning.

Give it a try and if you want to check your answer, write back. If you 
are having difficulties, let me know what you have done so far and 
I'll help you out.

Good luck.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/