let x ,y be lengths of sides of base , z be height of box
amount of cardboard used =surface area s=xy +2xz +2yz
volume v= x y z =27436
==>z=27436/xy
s=xy +2x*27436/xy +2y*27436/xy
s=xy +54872/y +54872/x
minimum amount ==>ds/dx=0 ,ds/dy =0
==>ds/dx= y-54872/x^2 =0 ,ds/dy= x-54872/y^2 =0
==> yx^2=54872 ,xy^2 =54872
xy^2 -yx^2 =0 ==>xy(y-x)=0 ==>y=x
yx^2=54872==>y=54872/x^2
xy^2 =54872==>x(54872/x^2)^2 =54872 ==>x^3 =54872 ==>x=38
==>y=38
xyz=27436 ==>z=27436 /xy ==>z=27436 /(38*38) ==>z=19
dimensions of box are 38x38x19
(largest side)38 (smallest side)19
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