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January 10th, 2012, 04:09 PM  #1 
Newbie Joined: Jan 2012 Posts: 9 Thanks: 0  Maximum Area of Circle and Square
Here's the question: A rope has length 10 cm, divided into 2 parts. A circle with radius is made from the first part and the second part is used to make a square with side . If sum of circle's area and square's area is maximum, calculate ! I get But, when I substitute again and recheck, my answer gives me minimum sum of area of circle + square. I do wrong? 
January 10th, 2012, 05:01 PM  #2  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond  Re: Maximum Area of Circle and Square Quote:
 
January 10th, 2012, 05:26 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,287 Thanks: 1023  Re: Maximum Area of Circle and Square
What's the area of a circle with circumference = 10 ? What's the area of a square with perimeter = 10 ? 
January 10th, 2012, 05:42 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Maximum Area of Circle and Square
We require the circumference of the circle plus the perimeter of the square to be 10 cm. With r and x in cm, we have: Solving for r, we find: Thus, the sum of the areas A is: We see we have a parabolic area function opening upward, so the minimum area will be at its vertex, and the maximum area will be either at x = 0 or x = 10: Since A(0) > A(2.5), we find the maximum area is obtained when all of the rope is used to make the circle, i.e., when x = 0. 
January 10th, 2012, 06:01 PM  #5 
Newbie Joined: Jan 2012 Posts: 9 Thanks: 0  Re: Maximum Area of Circle and Square
Okey, I got where is my wrong...I forget that Thanks all 
January 10th, 2012, 07:56 PM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,287 Thanks: 1023  Re: Maximum Area of Circle and Square Quote:
What's the area of a circle with circumference = 10 ? What's the area of a square with perimeter = 10 ?  
January 10th, 2012, 08:06 PM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Maximum Area of Circle and Square
Yes, it is well known that the circle encloses more area for a given perimeter than any polygon, like the sphere encloses more volume for a given surface area than any other solid. I wanted to show for certain that the maximum occurred at one of the endpoints. The problem would have been more fun to find the minimum. 

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