Expert: Paul Klarreich - 4/30/2008

Question
If 1200cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer
Questioner:   Rebecca
Category:  Calculus
Private:  No
 
Subject:  Maximum and Minimun Problem
Question:  If 1200cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
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Hi, Rebecca,

Determine your variables:

l, w, h are the dimensions.
V = volume of the box.

Write V in terms of the variables:

V = lwh.

Use other relations (constraints) to eliminate all but one variable:

w = l, because the base is a square.

Area = 1200 = w^2 + 4wh.

Solve that:  4wh = 1200 - w^2
   1200 - w^2
h = ----------
       4w

Now rewrite V in terms of w alone:

V = lwh = w^2h = w^2(1200 - w^2)/4w

V = w(1200 - w^2)/4 = (1200w - w^3)/4

OK, you're practically there.  Just diff, set it = 0, solve.