Question

A box with an open top has a length of x centimeters, width of y centimeters, height of z centimeters, and fixed volume of 125 cubic centimeters. The box is divided into two equal parts along its height. The bottom part is divided into two equal parts along its length. One of these parts is divided into two equal parts along its width. The sturdy material used for the base of the box costs $4 per square centimeter, and the material for the lateral sides of the exterior of the box costs $2 per square centimeter. The divider of least area inside the box is made of the same material as that of the base. The material for the divider whose area is the same as that of the base of the box costs $1 per square centimeter. The other divider is made of the same material as that of the lateral sides of the exterior of the box. Using our technique of Lagrange multipliers, determine the dimensions x, y, and - that minimize the material cost of this strange box.

Answer 1

divider I [/ N. n (6) n divider 2 dirides 3 2x 12/2 2x 조 t ng 2/2 (c) (d) final boxes 1x(b), IX(c) 2x(d) AS per question, divider L $ 1/cm² lateral divider 2 ) $2/cm² $41 cm? base, divider 3 xyz = 125 cost of total material in (b) = 2x/9*7)x2 + 2x(3x3)x2 + (axy)xt 2y2 + 2n2+ny in (c) (Yx2)x4 + 2x(***)*2 + (yx+)(2+2). ang +22+ 2yz in (d) = (3*3)*4 + 2x(yx )*2 + (xş)*2 + (x3)*4 xy + y2 + 2 day +242 + 3n2 for 2x(d)

Total cost (f(ny,2)) minimize 6azt say + byz (g(2, 2)) such that = 125 ayz 2,4, 2 so If = (5y +62, sn +62, 6utby) = (yz, az, ay) 5 + 6 = 1 y z egy If = 1 g 5y+62 =yz 5n+62 - luz 6n+by day . ad & + 2 = 1 from 2 6 & Tray y from & 3 = 1 3 (5y = 62] = 125 Now, nyz 25 yxyx sy y = 150 y= (150)" - 5.313 an 5.313 y = 4.428 an 2
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