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Unformatted text preview: L + 2 W + 2 H = 108 We are going to use Lagrange multipliers here. Let g ( L,W,H ) = L + 2 W + 2 H . We ﬁnd that ∇ V = h WH,LH,LW i and ∇ g = h 1 , 2 , 2 i . Computing ∇ V = λ ∇ g , we ﬁnd that we have four equations: WH = λ, LH = 2 λ, LW = 2 λ, L + 2 W + 2 H = 108 Using the second and third equations, along with the fact that the length of the box can’t be zero (that would be one lame box, after all), we ﬁnd that LH = LW , so H = W . Thus, going back to look at the ﬁrst and second equations, we know that LH = 2 λ = 2 WH = 2 H 2 , so L = 2 H . Now let’s plug all this jazz into the last equation and solve: 108 = L + 2 W + 2 H = 2 H + 2 H + 2 H = 6 H, so H = 108 / 6 = 18 Thus, W = 18 and L = 2 · 18 = 36. 1...
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This homework help was uploaded on 04/17/2008 for the course MATH 231 taught by Professor Bociu during the Spring '08 term at UVA.
 Spring '08
 BOCIU
 Math, Calculus

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