Let side of the square garden is a and radius of the circular garden is r.
We have to maximize a2+πr2
Subject to, 4a+2πr=100 i.e., 2a+πr=50
The Lagrange function L=(a2+πr2)+λ(2a+πr−50).
∴∂a∂L∂r∂L≡2a+2λ=0⇒λ=−a≡2πr+λπ=0⇒λ=−2r.
∴∴ and, 2(−λ)+π(−2λ)=50⇒λ(−2−2π)=50⇒λ=4+π−100a=4+π100r=4+π50
∴ The required fence for square =4+π400ft and for circular garden =4+π100πft.
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