Question

Question 10 A dosed rectangular box with a volume of 1626 is made from two kinds of materials. The top and bottom are made of material costing 48 cents per square foot and the sides from material costing 8 cents per square foot. Use Lagrange multipliers to find the dimensions of the box so that the cost of materials is minimized. Assume that w s I ft ft Question Attempts of I used SAVERS LATER MapleNet

Answer 1

Question 10: Leongth of box = L width = w height th & volume of bo v = (4) (W) (h). = 16 2743 # area of top and bottom of box A = Ab=ler * cost of material wed to make top and bottom of 4 (6= 48 cents/ft? box 4 = Q(Wh) + & total area of tides +alth) cost of material used to make sides of box (6= 8 cents/ft? Total c=24w (48) + kalh+2th) (6) of material wed cort c (h, w, h = 9 bult 16th + 16th (((, wh) = 9614+ 16uth & 16th let & the wy b) = wh- 162 =0 use Lagrange multipliers to minimize the cort of material used

Dc (Lush) = 100/ulh) (qow+ 16h, 962 +16h, I Gult 161) = Alwh.ch, w) 96W+16h = awh 961+16 h = 12h 16W+16l = 2ull 16 16 16 + 96 h + 16 L 96 T + W j L 0 Eg 0 and 90 & 0 = Equatona 96 + 6 + 2 y = 2 2 2 2 2 2 2 2 2 16 16 + 96 L w ñ h + a = 4 / W=L 96 h h 16 964 96 h=64 Eg ☺ and x=Lath=162 L(L) (61) = 162 613 = 162 13 - 162 6 23 = 27 L = 3 W=1-3 h=6= 6(3) = 18 L=3ft: W=3ft /h= 18ft