help me please. need this asap Use Lagrange Multipliers to find the solution to
ID: 2830938 • Letter: H
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help me please. need this asap
Use Lagrange Multipliers to find the solution to the word problem. (14.8) Find the maximum volume of a rectangular box without a lid which uses 108 square units of material. Find the minimum surface area of a right circular cylinder with volume equal to 54pi cubic units. (V = pir2h, SA = 2pir(r + h)) Find the area of the largest rectangle which has its base on the x-axis and fits in the triangle with vertices (-4,0), (0,8), (4,0). Find the highest and lowest points which lay on the curve of intersection for the cylinder x2 + y2 = 8 and the plane 2x + 2y + z = 16.Explanation / Answer
(a) Volume = lbh ; surface-area without lid = 2(lb+bh+lh) - lb
L = lbh + lambda*(108 + lb - 2(lb+bh+lh))
for optimal solution dL/dl = dL/db = dL/dh = dL/d(lambda) = 0
=>dL/dl = bh - lambda(2h+b) = 0
dL/db = lh - lambda*(2h + l) = 0
dL/dh = lb - lambda*(2l+2b) = 0
dL/d(lambda) = 108 + lb - 2(lb+lh+bh) = 0
=> solving above four equations we get =>
l=6;b=6;h=3 => volume= 6*6*3 = 108 cubic units
(b) L = 2*pi*r(r+h) + lambda(54*pi - pi*r*r*h)
doing the same as above in (a)=> we get=>
r = 3;h=6 => min surface area = 54*pi
(c),(d) similar setup
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