What is the least amount of material required to construct a square base box one of whose base is open and whose volume is 2700cm^3?

1 Answer
Feb 10, 2018

The least amount of material required to construct a square base box is
#A=923.387cm^2#

Explanation:

Let the side of the square base be #x#
Area of the square base is #x^2#
If the height of the box is #y#
Area of the lateral side is#xy#
Total lateral surface of the box is #4xy#
Including the base, material consumed will be
#A=4xy+x^2#
Volume of the box is #x^2y#
Given:
Volume of the box is #=2700cm^3#
Equating the volumes
#x^2y=2700#
Solving for y
#y=2700/x^2#
Substituting in the expression for area
#A=4x(2700/x^2)+x^2#
Simplifying
#A=4(2700/x)+x^2#

#A=10800/x+x^2#
Minimizing,
#(dA)/dx=-10800/x^2+2x#
By the principle of maxima and minima
Equating to zero
#(dA)/dx=0#
#-10800/x^2+2x=0#
For #x.ne0#
Multiplying throughout by #x^2#
#-10800+2x^3=0#
Simplifying

#2x^3=10800#

#x^3=10800/2#

#x=(10800/2)^(1/3#
#=5400^(1/3#
#x=17.54#

Substituting the value of x in the expression for Area

#A=10800/17.54+17.54^2#
#A=923.387cm^2#