SOLUTION: what is the least possible diagonal of a rectangle with a perimeter of 100
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Question 495663: what is the least possible diagonal of a rectangle with a perimeter of 100
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
If a and b are the side lengths with a+b = 50, we can find the minimum possible value of a diagonal several ways:
If you know power means, we have 
, equality occurs when a=b (square).
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The Cauchy-Schwarz inequality states that
(1+1) \ge (a+b)^2)
(2) \ge 50^2 \Rightarrow \sqrt{a^2+b^2} \ge 25\sqrt{2})
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We can say that ^2 = a^2 + b^2 + 2ab = d^2 + 2ab)
, the LHS is constant so we can maximize 2ab (twice the area of the rectangle) to minimize the diagonal d. We can show several ways the optimal case is a=b.
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All of these solutions yield 
as the minimal possible diagonal.
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