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An open-top box with a square base is to have a volume of 4 cubic feet. Find the dimensions of the box (in ft) that can be made with the smallest amount of material.

User Itsbalur
by
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1 Answer

8 votes

Answer:

The dimension of box=
2ft* 2ft* 1ft

Explanation:

We are given that

Volume of box=4 cubic feet

Let x be the side of square base and h be the height of box

Volume of box=
lbh=x^2h


4=x^2h


h=(4)/(x^2)

Now, surface area of box,A=
x^2+4xh


A=x^2+4x((4)/(x^2))=x^2+(16)/(x)


(dA)/(dx)=2x-(16)/(x^2)


(dA)/(dx)=0


2x-(16)/(x^2)=0


2x=(16)/(x^2)


x^3=8


x=2


(d^2A)/(dx^2)=2+(32)/(x^3)

Substitute x=2


(d^2A)/(dx^2)=2+(32)/(2^3)=2+4=6>0

Hence, the area of box is minimum at x=2

Therefore, side of square base,x=2 ft

Height of box,h=
(4)/(2^2)=1 ft

Hence, the dimension of box=
2ft* 2ft* 1ft

User Shenxian
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