Given that the volume of the aquarium is 20m^3.
Volume = Area of Base x height
Area of Base = Volume / height = 20/h
Given that the aquarium has a square base.
Area of square = l^2
Thus, the length of the base of the aquarium is [tex] \sqrt{area \ of \ base} = \sqrt{ \frac{20}{h} } [/tex]
The frame is to cover 8 sides with the length equal to the length of the base and 4 sides with the length of the height.
Thus, the total perimeter of the frame is given by [tex]8\sqrt{\frac{20}{h}}+4h= \sqrt{64\left(\frac{20}{h}\right)}+4h = \sqrt{\frac{1,280}{h}}+4h [/tex]
Area of the four side faces of the aquarium is 4 times the length of the base times the height = [tex]4\times\sqrt{ \frac{20}{h} }\times h=\sqrt{16\left(\frac{20}{h}\right)h^2}=\sqrt{320h}[/tex]
Total area to be covered by grass is the base and the four side faces and is given by [tex]\frac{20}{h}+\sqrt{320h}[/tex]
Cost of the entire metal frame = [tex]7\left(\sqrt{\frac{1,280}{h}}+4h\right)= \sqrt{49\left(\frac{1,280}{h}\right)}+7(4h) = \sqrt{\frac{62,720}{h}}+28h [/tex]
Cost of the entire grass = [tex]8\left(\frac{20}{h}+\sqrt{320h}\right)=\frac{160}{h}+\sqrt{64(320h)}=\frac{160}{h}+\sqrt{20,480h[/tex]
Therefore, total cost in terms of the height, h, is given by
[tex]C=\sqrt{\frac{62,720}{h}}+28h+\frac{160}{h}+\sqrt{20,480h[/tex]
