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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-




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
Anopenboxwithasquarebaseistobemadeoutofagivenquantityofacardboardofareac2squareunits.showthemaximumvolumeoftheboxc263cubicunits
Commented by PRITHWISH 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.
Lettheareaofthefloor=x2theareaoftheremaining4sides=c2x2theheightoftbebox=c2x24xThevol.V(x)=14(c2x2)xV(x)=14(c23x2)=0x2=c23V(x)=32x<0xVmax.=14(c2c23)c3=c363proved.
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)) ■
volumeofthebox=V(x)=(c2x)2xV(x)=(c2x)24(c2x)x=0(c2x){c2x4x}=0,x=16cVmax=(c13c)2.(16c)=(4c29)(c6)=2c327◼

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