Question

Your company is introducing a fruit drink packaged in an aluminum box with a square

base. Find the surface area of this box as a function of its dimension of its base, S, given
that volume of the box is 36 in. Graph this function and determine the dimensions that
produce a minimum surface area for this aluminum box.​

Answer 1

See explanation

Explanation:

Let x in be the base side length and y in be the height of the box. Since the base is a square, we have

`S=x^2\Rightarrow x=√(S)`

The volume of the box is

`V=S\cdot y\\ \\36=Sy\Rightarrow y=(36)/(S)`

The surface area of the box is

`SA=2x^2+4xy\\ \\SA(S)=2S+4\cdot √(S)\cdot (36)/(S)=2S+(144)/(√(S))`

The graph of the function SA(S) is shown in attached diagram.

Find the derivative of this function:

`SA'(S)=(2S+144S^{-(1)/(2)})'=2-(1)/(2)\cdot 144\cdot S^{-(1)/(2)-1}=2-(72)/(S√(S))`

Equate this derivative to 0:

`2-(72)/(S√(S))=0\\ \\2S√(S)=72\\ \\S√(S)=36\\ \\S^{(3)/(2)}=6^2\\ \\S=6^{(4)/(3)}`

So, the dimensions that produce a minimum surface area for this aluminum box are:

`x=\sqrt{6^{(4)/(3)}}=6^{(2)/(3)} \ in\\ \\y=\frac{6^2}{6^{(4)/(3)}}=6^{(2)/(3)}\ in.`