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 $$a3$$

Question

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 $$a3$$

Infinity Learn · Accepted Answer

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).

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 $a^{3}$

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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

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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 $a^{3}$