My Math Forum Maximum Area of Circle and Square

 Algebra Pre-Algebra and Basic Algebra Math Forum

 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 $r cm$ is made from the first part and the second part is used to make a square with side $x cm$. If sum of circle's area and square's area is maximum, calculate $x$! I get $x= \frac{10}{\Pi +4}$ 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:
 Originally Posted by zerostalk I get $x= \frac{10}{\Pi +4}$
How did you compute that?

 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: $2\pi r+4x=10$ Solving for r, we find: $r=\frac{10-4x}{2\pi}=\frac{5-2x}{\pi}$ Thus, the sum of the areas A is: $A(x)=\pi$$\frac{5-2x}{\pi}$$^2+x^2=\frac{$$\pi+4$$x^2-20x+25}{\pi}$ 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: $A(0)=\frac{25}{\pi}$ $A$$\frac{10}{4}$$=\frac{25}{4}$ 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 $0 \leq a,b\leq 10$ Thanks all
January 10th, 2012, 07:56 PM   #6
Math Team

Joined: Oct 2011

Posts: 14,597
Thanks: 1038

Re: Maximum Area of Circle and Square

Quote:
 Originally Posted by MarkFL we find the maximum area is obtained when all of the rope is used to make the circle, i.e., when x = 0.
Of course; area circle circumference x > area square perimeter x; which is why I posted:
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

,

,

,

,

,

,

# calculating maximum of area of circle and a square

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post yeoky Algebra 4 May 3rd, 2014 01:06 AM PlzzHelp Algebra 2 December 5th, 2013 09:45 AM gus Algebra 1 April 17th, 2011 04:25 PM captainglyde Algebra 1 February 19th, 2008 08:55 AM brunojo Algebra 2 November 16th, 2007 10:30 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top