If 2400 square centimeters of material is available to make a box with a square base and an open top, how do you find the largest possible volume of the box?

1 Answer
Gió
Nov 3, 2015

The dimension of the box I found are:
#28.3xx28.3xx14.1cm# for a volume of #11,292.5cm^3#:

Explanation:

The surface #S# of a box with open top is the sum of the surfaces of a square (base, of side #a#) and 4 rectangles (base #a# and height #h#):
#S=a^2+4a*h# (1)
while the volume #V# will be the area of the base times the height, or:
#V=a^2*h# (2)

We get #h=(S-a^2)/(4a)# from (1) put it into (2):

#V=a^2(S-a^2)/(4a)=1/(4a)(Sa^2-a^4)=S/4a-a^3/4#
maximize this volume deriving with respect to #a# and setting it equal to zero:
#(dV)/(da)=S/4-3/4a^2=0#
#3/4a^2=S/4#
#a^2=S/3#
#a=+-sqrt(2400/3)=+-28.3cm#

We use #a=+28.3cm# that in (1) gives us:
#h=(2400-28.3^2)/(4*28.3)=14.1cm#

Giving a volume of:
#V=a^2*h=11,292.5cm^3#

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