What is the maximum volume of an open box with a square base whose surface area (not including the top) is #27# #i n ^2#?

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Answer 1 · 2016-10-21T15:22:33

Let #l# be the length of the square base and #h# be the height. Then the surface area, #S.A#, is given by:

#S.A = l^2 + 2lh + 2lh = 27#

#l^2 + 4lh = 27#

Solve for one of the variables.

#4lh = 27 - l^2#

#h = (27 - l^2)/(4l)#

The formula for volume of the box is #V = l xx l xx h#.

#V = l(l)(27 - l^2)/(4l)#

#V = (27l - l^3)/4#

You can determine the maximum value of this function using graphing calculator.

For the maximum, you should get a maximum volume of #13.5" in"^3#. A length of #3# inches would give this maximum.

Hopefully this helps!