Assume that S=surface area=1. Use LaGrange multipliers to find box of maximal volume. Hence, maximize V(x,y,x)=x,y,z subject to the constraint 2xy+2xz+2yz=1 for a box:
(a) with top
(b) WITHOUT top

Question

Assume that S=surface area=1. Use LaGrange multipliers to find box of maximal volume.  Hence, maximize V(x,y,x)=x,y,z subject to the constraint 2xy+2xz+2yz=1 for a box:
(a) with top
(b) WITHOUT top

anonymous · Answer

Surface Area Formula(with top) ---> constraint(G) 2 L W + 2 W H + 2 L H =1 Volume formula (To be maximize ---> F) L W H Find Gradient of F and G; w.r.t (l , w , h) F= G=<2W+2H, 2L + 2H, 2W + 2 L> $$F=\lambda G$$ $$WH=\lambda(2W+2H)$$ $$LH=\lambda(2L + 2H)$$ $$LW=\lambda(2W + 2 L)$$ $$2 L W + 2 W H + 2 L H =1$$ Four equations with four variable