Respuesta :
The surface area of the box in terms of x is [tex]\frac{x^{3} +476656}{x}[/tex] and the surface area is minimum at x=62cm.
The quantity of a rectangular box is given by using V=duration × width × peak.
The duration and width of the base of the rectangular container is x.
allow height of the container =h
Volume = 119164 cubic cm
Volume of box=length × width ×height
[tex]or,x^{2} h=119164\\or,h=\frac{119164}{x^{2} }[/tex]
Surface area of the box=base area+4(area of 1 wall)
or, surface area=[tex]x^{2} +4xh[/tex]
Let us denote the surface area as a function of x.
[tex]A(x)=x^{2} +\frac{4\times(119164)}{x}[/tex]
Now let us find the first derivative of A(x).
[tex]A'(x)=2x-\frac{476656}{x^{2} }[/tex]
At[tex]A'(x)=0\\[/tex]
[tex]2x-\frac{476656}{x^{2} }=0\\or, x=62[/tex]
Now A(x) is minimum for x, at A'(x)=0.
Hence the surface area will be [tex]\frac{x^{3} +476656}{x}[/tex] which will be minimum at x=62.
To learn more about the volume of a solid figure:
https://brainly.com/question/11168779
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