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(A) For the cylinder to float in equilibrium,the weight of the cylinder must be equal to the total buoyant force exerted by the two liquids.Let $A_c =

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$5/4 d$

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(A) For the cylinder to float in equilibrium,the weight of the cylinder must be equal to the total buoyant force exerted by the two liquids.Let $A_c = A/5$ be the cross-sectional area of the cylinder.The length of the cylinder submerged in the upper liquid (density $d$) is $3L/4$.The length of the cylinder submerged in the lower liquid (density $2d$) is $L/4$.Weight of the cylinder = $V_{total} 	imes D 	imes g = (A_c 	imes L) 	imes D 	imes g = (A/5) 	imes L 	imes D 	imes g$.Buoyant force = (Weight of upper liquid displaced) + (Weight of lower liquid displaced)Buoyant force = $(A_c 	imes 3L/4) 	imes d 	imes g + (A_c 	imes L/4) 	imes 2d 	imes g$Buoyant force = $(A/5) 	imes (3L/4) 	imes d 	imes g + (A/5) 	imes (L/4) 	imes 2d 	imes g$Equating weight and buoyant force:$(A/5) 	imes L 	imes D 	imes g = (A/5) 	imes g 	imes [ (3L/4) 	imes d + (L/4) 	imes 2d ]$$L 	imes D = (3L/4) 	imes d + (2L/4) 	imes d$$L 	imes D = (5L/4) 	imes d$$D = 5/4 d$

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