(5) Note: This is the same bonus problem as Lesson 12. This time, you have to use the method of Lagrange Multipliers to solve it. A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width1 height) with the smallest possible surface area. V = xyz cy= = 500 Tank open at top Surface area = length * width + 2 * height(length + width) S = xy + 2z(xty) f(x, y) = xy+ 1000 1000 + y fx = y - 1000 -ify = x- 1000 Critical value : xy = 1000; cy' = 1000 xy(x - y) =0 x = 0, y = 0, x =y Critical point : (10, 10) 2000 13 ifyy = 2000 13 i fry = 1 D(x, y) = fxifyy - (fry)2 D(10, 10) : 4-1 =3 >0 for > 0 So, Surface area minimum (10, 10) Length = 10 Width = 10 Height = 5 Sur facemin = 10* 10 + 2 * 5(10 + 10) Sur face min 300 f+2