A small garden
A gardener wants to create a closed fenced field. However, he only has 4 different fencing lengths: one of 1m, one of 2m, one of 3m and one of 4m. If the maximum area that can be fenced can be represented as a b , where a and b are positive integers and b is squarefree, find a + b .
The answer is 8.
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1 solution
Can you prove your first claim?
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Let me do it XD
Bretschneider's formula states that the area of a quadrilateral is given by ( s − a ) ( s − b ) ( s − c ) ( s − d ) − a b c d ⋅ c o s 2 ( 2 A + C ) where A and C are opposite angles of the quadrilateral. Area is maximised when 2 A + C = 9 0 , which means that the quadrilateral is cyclic.
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Nice! Knocked it out of the park in just a couple of lines.
I've added the proof to my problem solution. Thanks.
Same. Max. with cyclic quadrilateral. Also max with equilateral triangle. Max area given perimeter is square.
If the fence panels could be bent into circular arcs of radius = 1.591549m , it is possible to enclose 7.9577 square meters of area by using a circle. That's quite an improvement over the convex quadrilateral!!