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)) ■