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Homework answers / question archive / Find the maximum and minimum volumes of a rectangular box whose surface area equals 8000 square cm and whose edge length (sum of lengths of all edges) is 480 cm

Find the maximum and minimum volumes of a rectangular box whose surface area equals 8000 square cm and whose edge length (sum of lengths of all edges) is 480 cm

Math

Find the maximum and minimum volumes of a rectangular box whose surface area equals 8000 square cm and whose edge length (sum of lengths of all edges) is 480 cm. Hint: It can be deduced that the box is not a cube, so if x, y, and z are the lengths of the sides, you may want to let x represent a side with

and

. Maximum value is ,
occuring at (, ,). Minimum value is ,
occuring at (, ,).

 

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Please see the attached file.

Solution
f(x, y, z) = xyz, volume
g(x, y, z) = 2xy + 2yz + 2xz - 8000
h(x, y, z) = 4x + 4 y + 4 z - 480

We have equations (using Lagrange multipliers):
f(x, y, z) = 0
h(x, y, z) = 0
grad_f = L * grad_g + M * grad_h

v = [x, y, z],
grad_f = grad (f, v)
grad_h = grad (h, v)

grad_f = [yz, xz, xy]
grad_g = [2y + 2z, 2x + 2z, 2x + 2y]
grad_h = [4, 4, 4]

System of equations:

yz - L (2y + 2z) - 4 M = 0
xz - L (2x + 2z) - 4 M = 0
xy - L (2x + 2y) - 4 M = 0
2 xy + 2 yz + 2 xz - 8000 = 0
4 x + 4 y + 4 z - 480 = 0

The solution of the system of equations gives the above solution.

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