Question Number 100341 by peter frank last updated on 26/Jun/20

An open box with a square base is to be made out of a given quantity of a cardboard of area c^2 square units.show the maximum volume of the box (c^2 /(6(√3))) cubic units

Commented byPRITHWISH SEN 2 last updated on 26/Jun/20

Let the area of the floor = x^2 the area of the remaining 4 sides = c^2 −x^2 ∴ the height of tbe box = ((c^2 −x^2 )/(4x)) ∴ The vol. V(x)=(1/4)(c^2 −x^2 )x V^′ (x)=(1/4)(c^2 −3x^2 )=0⇒x^2 =(c^2 /3) V^(′′) (x)=−(3/2)x<0 ∀x ∴V_(max.) = (1/4)(c^2 −(c^2 /3))(c/(√3)) = (c^3 /(6(√3))) proved.

Answered by bobhans last updated on 26/Jun/20

volume of the box = V(x)= (c−2x)^2 x V′(x)= (c−2x)^2 −4(c−2x)x = 0 (c−2x){c−2x−4x} = 0 , x = (1/6)c V_(max) = (c−(1/3)c)^2 .((1/6)c) = (((4c^2 )/9))((c/6)) = ((2c^3 )/(27)) ■