What dimensions will result in a box with the largest possible volume if an open rectangular box with square base is to be made from #48 ft^2# of material?

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Answer 1 · 2015-02-22T12:51:38

The base will be # 4 xx 4# and the height will be #2# (all numbers in feet)

Let #b# be the length and the width of the base (length and width are the same since the base is square).

Let #h# be the height of the box.

The surface area of the box is
#base + height xx perimeter#
#= b^2 + 4bh = 48#
From which we can determine:
#h = (48 - b^2)/(4b)#

The Volume of the box:
# V(b) = b^2h = b^2 * ((48 - b^2)/(4b))#
# = 12b - (3b^3)/4#

The Volume is a maximum when #( d V(b))/ (db) = 0#

#(d V(b))/(db) = 12 - (3 b^2)/4 = 0#

#b = +-4# (only #+4# is not extraneous)

Plugging this back into the formula
#h = (48 - b^2)/(4b)#
we get
#h = 2#