68 inches of string
Divide the 68 inch string into two parts. With one
part make a square figure, the other make a circle.
What are the length of two parts of string that will
maximize the sum of the figure's area.
Warning: This one is pretty messy.
part make a square figure, the other make a circle.
What are the length of two parts of string that will
maximize the sum of the figure's area.
Warning: This one is pretty messy.
Labels: mathschallenge
posted by Ragknot at
3:23 PM
4 Comments:
Not messy at all!
Combined area of the two figures is maximised when all of the string (bar the smallest part that can be cut from it) goes to making the circle of just under 368 square inches. The rest goes to make a nominal square of minimum size.
We have two lengths: x and 68-x
where x is between 0 and 68 inclusive
lets say x is for the circle and 68-x will be the square
since x is the perimeter of the circle we can solve for the radius and then the area
C = 2pi*r
x = 2pi*r
r = x/2pi
A = pi(x/2pi)^2
A = x²/4pi
we can do the same for the square
p = 4w
68-x = 4w
w = 17-x/4
A = (17-x/4)²
adding the two areas we get
A = x²/4pi + (17-x/4)²
to find the maximum area we take the derivative and solve for when dA/dx=0
A = x²/4pi + (17-x/4)²
0 = x/2pi - (17-x/4)/2
x = 68/(1+pi)
But Wait! this is a local minimum. This can be verified by doing the second derivative test.
A" = 1/2pi+1/8
So what now? Well consider this: if x=68/(1+pi) is a local minimum then on either side of this point the area will get larger. So the further away you get from this point the larger the area. Let's takes the extreme cases when x=0 and x=68. Note that this means that we are not cutting the string and using the whole string to make either a square or a circle.
A[square] = (68/4)²
A[square] = 289
A[circle] = 68²/4pi
A[circle] = 367.9662
Therefore, to get the maximum area we don't cut anything and use the whole string to make a circle.
Sam,
That was a brilliant prove that between a square and a circle of equal perimeters, the circle has more area.
It was interesting reading.
But unfortunetly, few others recognized that fact.
Sam,
that was most definitely a brilliant set-up and i found it very interesting! reading it a couple times it actually made sense for me
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