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January 10th, 2012, 05:09 PM   #1
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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?
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January 10th, 2012, 06:01 PM   #2
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Re: Maximum Area of Circle and Square

Quote:
Originally Posted by zerostalk
I get
How did you compute that?
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January 10th, 2012, 06:26 PM   #3
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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 ?
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January 10th, 2012, 06:42 PM   #4
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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.
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January 10th, 2012, 07:01 PM   #5
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Re: Maximum Area of Circle and Square

Okey, I got where is my wrong...I forget that

Thanks all
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January 10th, 2012, 08:56 PM   #6
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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 ?
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January 10th, 2012, 09:06 PM   #7
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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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