Answer:
[tex]V=74,052\ cm^3[/tex]
Step-by-step explanation:
we know that
The maximum volume for a rectangular prism is when the base is a square
The girth is the distance around the package perpendicular to the length
see the attached figure to better understand the problem
Let
x ----> the length of the square base of the prism
h ---> the height of the prim
we have that
[tex]4x+h=2[/tex]
[tex]h=2-4x[/tex] -----> equation A
The volume of a rectangular prism is given by
[tex]V=Bh[/tex]
where
B is the area of the base
h is the height of the prism
we have
[tex]B=x^2[/tex]
so
[tex]V=x^2h[/tex] ----> equation B
substitute equation A in equation B
[tex]V=x^2(2-4x)[/tex]
[tex]V=-4x^3+2x^2[/tex]
Find the first derivative of the function
[tex]\frac{dV}{dx}=-12x^2+4x[/tex]
equate to zero
[tex]-12x^2+4x=0\\-12x(x-\frac{1}{3})=0[/tex]
[tex]x=0.33\ m[/tex] ---->[tex]x=33\ cm[/tex]
[tex]h=200-4(33)=68\ cm[/tex]
therefore
The greatest volume is equal to
[tex]V=(33^2)(68)=74,052\ cm^3[/tex]
