SOLUTION: a manufacturer wants to design a box having a base where the length is twice the width. the surface area must be exactly 300 square inches. what dimensions will produce a maximum v

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Question 1082325: a manufacturer wants to design a box having a base where the length is twice the width. the surface area must be exactly 300 square inches. what dimensions will produce a maximum volume?
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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So you know that,
2X%5E2%2B2%282X%29Z%2B2XZ=300
2X%5E2%2B4XZ%2B2XZ=300
2X%5E2%2B6XZ=300
X%5E2%2B3XZ=150
Z=%28150-X%5E2%29%2F%283X%29
The volume of the box is,
V=2X%28X%29%28Z%29
V=2X%5E2%28%28150-X%5E2%29%2F%283X%29%29
V=%282%2F3%29X%28150-X%5E2%29
You can maximize by taking the derivative and setting it equal to zero,
dV%2FdX=%282%2F3%29X%28-2X%29%2B%282%2F3%29%28150-X%5E2%29
dV%2FdX=-2%28X%5E2-50%29
So,
X%5E2-50=0
X%5E2=50
X=sqrt%2850%29
X=5sqrt%282%29
So then,
2X=10sqrt%282%29
and
Z=100%2F%283sqrt%2850%29%29
Z=%2810%2F3%29sqrt%282%29

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor forgot to put a top on the box.
The box when flattened out looks like this, drawn
to scale:



He did it correctly for a box without a top.
Follow his procedure but add the surface area
of the top of the box.

Answer for the box with a top, X = 5 in., 2X = 10 in., Z = 10 in.

It turned out that 2X and Z both were equal to 10, but
of course, you can't use that fact when solving.

Edwin