Given rectangular solid have dimensions,
[tex]x=7.5cm, y=7.5cm[/tex]
And maximum volume is, [tex]421.875cm^{3}[/tex]
Given that A rectangular solid with a square base. It means that rectangular solid has two square surfaces and 4 rectangular surfaces.
Since, x represent the length of the sides of the square base and y represent the height.
Therefore, surface area of rectangular solid is, = [tex]2x^{2} +4xy[/tex]
And surface are a is given that 337.5 [tex]cm^{2}[/tex]
So , [tex]2x^{2} +4xy=337.5\\\\x^{2} +2xy=168.75\\\\y=\frac{168.75-x^{2} }{2x} .........(1)[/tex]
Volume of rectangular solid, [tex]V=x^{2} y[/tex]
Substituting the value of y from equation 1 in volume expression.
[tex]V=x^{2}*\frac{ (168.75-x^{2} )}{2x} \\\\V=\frac{1}{2}(168.75x-x^{3} )[/tex]
For maximum volume , differentiate above expression with respect to x and equate with zero.
[tex]\frac{dV}{dx}=\frac{1}{2} (168.75-3x^{2} ) =0\\\\x^{2} =\frac{168.75}{3}=56.25\\\\x=\sqrt{56.25}=7.5cm[/tex]
Substituting the value of x in equation 1.
We get, [tex]y=\frac{168.75-(7.5)^{2} }{2*7.5} \\\\y=7.5cm[/tex]
Maximum volume , V = [tex](7.5)^{3}=421.875cm^{3}[/tex]
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