An open topped rectangular box with a square base has a surface area of 10,000 #cm^2#. What are the dimensions of the box that make its volume a maximum and what is this maximum volume?

1 Answer
Alan N.
Jun 11, 2017

Base dimension #approx 57.74 cm#

Height dimension #approx 28.87cm#

#V_"max" =1,000,000sqrt3/18 approx 96,225.04# #cm^3#

Explanation:

Let the side on the square base be #x# cm and the height of the rectangular box be #y# cm.

Surface area of the box #S =# Base area + 4 side areas

#S= x^2+4xy#

Volume of the box #V = x^2y#

We are told that #S=10,000#

#:. x^2+4xy=10000#

#y = (10000-x^2)/(4x)#

Replacing for #y# in equation for #V#

#V=x^2* (10000-x^2)/(4x) = (x(10000-x^2))/4#

#= 1/4(10000x-x^3)#

#(dV)/dx = 1/4(10000-3x^2)#

For maximum or minimum #V#: # (dV)/dx=0#

I.e. #1/4(10000-3x^2)=0#

#3x^2 = 10000#

#x = +- sqrt(10000/3)= +- 100/sqrt3 =+- (100sqrt3)/3#

#x>0 -> x=(100sqrt3)/3#

#(d^2V)/dx^2 = (-6x)/4 < 0 # for #x>0#

#:. V_"max" = V((100sqrt3)/3)#

#= 1/4((10000xx100sqrt3)/3 - ((100sqrt3)/3)^3)#

#=1000000/4(sqrt3/3-(sqrt3)^3/27)#

#=1000000*sqrt3/12(1-3/9)#

#=1000000*sqrt3/12(2/3)#

#=1000000*sqrt3/18#

#approx 96,225.04# #cm^3#

We are asked for the dimensions of the box:

Base dimension #= x=(100sqrt3)/3 approx 57.74 cm#

Height dimension #= y = (10000-(100sqrt3/3)^2)/(4(100sqrt3/3))#

#=(10000(1-3/9))/(4xx100xx(sqrt3/3))#

#=(100(2/3))/(4xxsqrt3/3) #

#= 100xx2/(4xxsqrt3) =50/sqrt3 = (50sqrt3)/3#

#approx 28.87cm#

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