A solid rectangular brick is to be made from 1 cubic feet of clay. The length of the brick must be 3 times as long as its width. The width of brick for which it will have minimum surface area is  then a, is a3

Let width of a rectangular brick is  xlength of a rectangular brick is  3xheight of a rectangular brick is  h Volume of the rectangular brick, (x)(3x)(h)=1⇒3x2h=1⇒h=13x2  Surface area of the brick S=(x)(3x)+(3x)(h)+(h)(x)⇒S=6x2+8hx⇒S(x)=6x2+8x13x2⇒S=6x2+83x⇒S′(x)=12x−83x2 For maxima or minima S′(x)=0⇒12x−83x2=0⇒x3=29⇒x=2913S′′2913=12+829=48>0S′′(x)=12+8x3∴ surface area is minimum when width x=2913=a⇒a3=29Therefore, the correct answer is (2).