
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 7,958 Thanks: 1146 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: 14,597 Thanks: 1038  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: 521 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: 14,597 Thanks: 1038  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: 521 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. 

Tags 
area, circle, maximum, square 
Search tags for this page 
maximum area of a circle,minimum area of circle,How to find the maximum area of a circle,How to find the maximum or minimum area of a circle,calculate maximum area of a circle,calculating maximum area of a circle,calculating maximum of area of circle and a square
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Area enclosed by a semicircle and a quarter circle  yeoky  Algebra  4  May 3rd, 2014 01:06 AM 
Find the maximum area  PlzzHelp  Algebra  2  December 5th, 2013 09:45 AM 
Area of square with convex parts  area of circle  gus  Algebra  1  April 17th, 2011 04:25 PM 
area of eq. triangle vs. area of square  captainglyde  Algebra  1  February 19th, 2008 08:55 AM 
Maximum and minimum area for ngon  brunojo  Algebra  2  November 16th, 2007 10:30 AM 