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Sagot :
The minimum surface area for the rectangular container is [tex]588cm^{2}[/tex].
How to find the surface area?
A solid object's surface area is a measurement of the overall space that the object's surface takes up. Compared to the definition of the arc length of a one-dimensional curve or the definition of the surface area for polyhedra (i.e., objects with flat polygonal faces), where the surface area is equal to the sum of the areas of its faces, the mathematical definition of surface area in the presence of curved surfaces is much more complex.
Let s be the side of the square base and h be the height.
Surface Area=[tex]s^{2}+4sh[/tex]
Volume=[tex]s^{2}h[/tex]
According to the question,
[tex]s^{2}h=1372\\ h=\frac{1372}{s^{2} }[/tex]
So, surface area=[tex]s^{2} +4s(\frac{1372}{s^{2} })[/tex]
=[tex]s^{2}+\frac{5488}{s}[/tex]
Differentiate with respect to s,
Surface area=[tex]2s-\frac{5488}{s^{2} }[/tex]
Now, [tex]2s-\frac{5488}{s^{2} }=0[/tex]
[tex]2s=\frac{5488}{s^{2} } \\2s^{3}=5488\\ s^{3}=2744\\ s=14[/tex]
Find the value of h from the volume.
[tex]14*14*h=1372\\h=\frac{1372}{14*14}\\ h=7[/tex]
Thus, the minimum surface area=[tex]14^{2}+4*14*7[/tex]
=[tex]588cm^{2}[/tex]
Learn more about the Surface area here:
https://brainly.com/question/20771646
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