Mowing Half a Lawn

Date: 03/31/2002 at 11:45:04
From: Jessica Pion
Subject: Word Problem

Chris and Lee are each responsible for mowing half of their 50-foot 
by 90-foot rectangular back yard. Chris starts mowing at an outside 
corner, gradually working toward the middle by mowing around the 
outside edges. If the mower cuts a 3-foot wide path, during which trip 
around the yard should Chris stop and Lee start mowing?

I understand that the sides of the lawn get shorter every time Chris 
goes around. I don't understand what formula I would use to figure 
how much the lawn decreases. 

Thank you,
Jessica Pion

Date: 03/31/2002 at 18:37:08
From: Doctor Ian
Subject: Re: Word Problem

Hi Jessica,

 Let's draw a picture to help us visualize what's going on:  
           
                90
   +----------------------------------+
   |                                  |
   |  +----------------------------+  |
   |  |                            |  |
   |  |  +----------------------+  |  |
   |  |  |                      |  |  |
   |  |  |                      |  |  | 50
   |  |  |                      |  |  |
   |  |  +----------------------+  |  |
   |  |                            |  |
   |  +----------------------------+  |
   |                                  |
   +----------------------------------+ 

Each of the strips is 3 feet wide, so after n trips around the yard, 
thesituation looks like this: 

                90
   +----------------------------------+
   |                                  |
   |  +----------------------------+  |
   |  |                            |  |
   |  |  +----------------------+  |  |
   |  |  |      90 - 6n         |  |  |
   |  |  |              50 - 6n |  |  | 50
   |  |  |                      |  |  |
   |  |  +----------------------+  |  |
   |  |                            |  |
   |  +----------------------------+  |
   |                                  |
   +----------------------------------+ 

So after some number of trips, the amount of yard left for Lois to 
cut, 

  (90 - 6n)(50 - 6n) 

is going to be less than half of the total amount of yard, 

  (1/2)(90)(50) 

which means that you want to find n such that 

  (90 - 6n)(50 - 6n) <  (1/2)(90)(50)

Does this help? 

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