SOLUTION: There is a box with a square base, and a volume of 32000cm^3, with an open top, what is the least material used, and what are the dimensions?

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Question 30833: There is a box with a square base, and a volume of 32000cm^3, with an open top, what is the least material used, and what are the dimensions?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let a side of the square base = x
Let the height of the box = y
The volume, V =
V+=+32%2A10%5E3
32%2A10%5E3+=+x%5E2+%2A+y
The dimensions of the materials = 4*x*y for the sides
and x^2 for the base
M = materials
M+=+4%2Ax%2Ay+%2B+x%5E2
from above
y+=+32%2A10%5E3+%2F+x%5E2
M+=+4+%2A+x+%2A+%2832%2A10%5E3%2F+x%5E2%29+%2B+x%5E2
M+=+4+%2A+%2832%2A10%5E3%2F+x%29+%2B+x%5E2
find M'= MP the dirivative of M
MP+=+32%2A10%5E3%2A+%28-4%2Ax%5E%28-2%29%29+%2B+2+%2A+x
set MP = 0 to find a minimum
0+=+32%2A10%5E3%2A+%28-4%2Ax%5E%28-2%29%29+%2B+2+%2A+x
%2832+%2A+10%5E3+%2A+4%29+%2F+x%5E2+=+2%2A+x
multiply both sides by x^2
x%5E3+=+4+%2A+32+%2A+10%5E3
x%5E3+=+64+%2A+10%5E3
x+=+40
The base is 40X40 cm
y+=+32%2A10%5E3+%2F+x%5E2
y+=+32%2A10%5E3+%2F+40%5E2
y+=+32+%2A+10%5E3+%2F+16+%2A+10%5E2
y+=+20
The height is 20 cm