Answer :
The dimensions of the box that minimize total cost are height h =3.17m, width, w and length, l=9.52m are equal to .
How to find the height of the box ?
Let h is the height of the box.
Given Volume of the box is 216 m³ with no top.
And the cost of the bottom is 40USD/m².
For the sides, it is 30USD/m².
The bottom of the box is square, so l = w.
So Volume
[tex]V=hl^{2}[/tex]
[tex]216=hl^{2}[/tex]
[tex]h=\frac{216}{l^{2} }[/tex]
Now Surface Area of the box without a top is
S = 4hl + l²
So,
cost = 4 × 30hl +40l²
cost = 120 hl + 40 l²
Putting h ,
cost = 120 × [tex](\frac{216}{l^{2} })l + 40l^{2}[/tex]
cost = [tex]\frac{25960}{l^{2} }+40l^{2}[/tex]
To find minimum cost, derivate the by using the calculator at
[tex]l=3\sqrt[3]{4}[/tex]
l=4.762203 m
[tex]h=\frac{216}{4.762203^{2} }[/tex]
h = 9.5244069354 m
Hence the dimensions of the box are
l=w
=9.52 m
and height ,h=3.17 m
Know more about the volume of the box here:
brainly.com/question/11168779
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