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Cleaning the Ice

Date: 09/09/97 at 22:31:11
From: Lauren Jones
Subject: Area

Dear Dr. Math,

Here is my question:

Venski is fascinated by the ice cleaning machine as it scrapes 
the ice between periods of a hockey game. The hockey rink is a 
rectangle, 120 ft. by 60 ft. The scraper cleans a 4 ft wide strip.  
If the machine starts at one corner and moves around and around the 
rink toward the center, on which trip around will it have half the 
area of the rink clean?

I have drawn a picture and divided and multiplied.  I am having 
trouble comprehending all of the numbers.
Respectfully yours,
Lauren Jones

Date: 09/10/97 at 16:48:30
From: Doctor Barney
Subject: Re: Area

I cannot think of any shortcut to solve this problem, so let's just 
figure out what half the area of the rink is and then add up how much 
area the "Zamboni" cleans on each trip around the rink.  

A = 120 x 60 = 7,200 square feet, so half of that area is 3,600 sq.ft.

first time around  = 4 x (120+56+116+52) = 1,376
second time around = 4 x (112+48+108+44) = 1,248  
                             total cleaned so far 2,624
third time around  = 4 x (104+40+100+36) = 1,120  
                             total cleaned so far 3,744

So, near the end of the third trip around the rink half of the area 
will have been cleaned.  

Notice that each time the machine turns a corner you need to subtract 
four feet from the next side because it has already cleaned the first 
four feet of that side where it turns the corner.  

For more fun try this one: How many trips around will the machine have 
made by the time all of the area is cleaned?  

-Doctor Barney,  The Math Forum
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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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