Question

3) You are constructing a 12 cubic feet box that is open at the top. The material used to construct the bottom costs $3 per square foot and the material used to construct the sides is $1 per square foot. What dimensions should you use to minimize the cost of the materials? Set up and solve using the LaGrange method.

Answer 1

2 3 the volume of box is xyz= - 314,4,2) = 132 for bottom the bottom x lost per totul Cust area of 34x for side the total costo mit 2 cua उ N Wall x cost per unit z22y +222 the totull kayt to constrout box b 3) hawa 2) IL ) ） A (X2)-0 8 f 19,9,2) - 3y x+ 2xy + 2x2 using lungnage multipler we y f (x, y, z) = 799 gex,y,z) 3y +22=0 (92) 30+22= Ry+2 acxy) xyzz12 syX+227 3xy + tazy (mentity in 9 by i 226 - 2zy zo 22 (x-3) 20 ::270 1224 similgy mutiplij in ③ by y 4 ③ by ? 3 xy + 224 py Ź+212 B x / 3y [3y = 22 4 in @ by y. 22) 26 So we hay xzy = 2z =k. 3

ha... Subsituty in 7 we k3 2 Koko 3k Ek I RXL 3 28 Ik za k3 - 122 373 ) Hence Cast T22 ,12= = 3 ?(3)(3) +()(3) +372)46) Rtrt 12 36 minimum Cost