How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of $8, 788 {(c m)}^{3}$ $8,788 (cm)^3$ ?

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How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of $8, 788 {(c m)}^{3}$ $8,788 (cm)^3$ ?

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Answer 1 · 2015-02-27T15:14:40

You set $x$ $x$ as being the sides, and $h$ $h$ for the height.

The box will have a square bottom.
Then the amount of cardboard used will be:
For the bottom: $x \cdot x = x^{2}$ $x*x=x^2$
For the sides: $x \cdot h \cdot 4$ $x*h*4$ (sides) $= 4 x h$ $=4xh$

Total area : $A = x^{2} + 4 x h$ $A=x^2+4xh$

The volume of the box= $x \cdot x \cdot h = 8788$ $x*x*h=8788$ from which we can conclude that $h = \frac{8788}{x^{2}}$ $h=8788/x^2$

Substituting that into the formula for the area $A$ $A$ , we get:

$A = x^{2} + 4 x \cdot (\frac{8788}{x^{2}}) = x^{2} + \frac{35152}{x}$ $A=x^2+4x*(8788/x^2)=x^2+35152/x$

To find the minimum, we have to differentiate and set to $0$ $0$

$A'=2x-35152/x^2=0->2x=35152/x^2$ multiply by $x^2$

$2x^3=35152->x^3=17576->x=root 3 17576=26$
Substitute: $h=8788//26^2=13$

Answer :
The sides will be $26cm$ and the height will be $13cm$

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How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of $8, 788 {(c m)}^{3}$ $8,788 (cm)^3$ ?

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