(Solved): 51. Find the volume of the largest rectangular box in the first octant with three faces in the coor ...

51. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane \( x+2 y+3 z=6 \). 52. Find the dimensions of the rectangular box with largest volume if the total surface area is given as \( 64 \mathrm{~cm}^{2} \). 53. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant \( c \). 54. The base of an aquarium with given volume \( V \) is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.