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
![\sqrt{area \ of \ base} = \sqrt{ \frac{20}{h} }](https://tex.z-dn.net/?f=%20%5Csqrt%7Barea%20%5C%20of%20%5C%20base%7D%20%3D%20%5Csqrt%7B%20%5Cfrac%7B20%7D%7Bh%7D%20%7D%20)
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
![8\sqrt{\frac{20}{h}}+4h= \sqrt{64\left(\frac{20}{h}\right)}+4h = \sqrt{\frac{1,280}{h}}+4h](https://tex.z-dn.net/?f=8%5Csqrt%7B%5Cfrac%7B20%7D%7Bh%7D%7D%2B4h%3D%20%5Csqrt%7B64%5Cleft%28%5Cfrac%7B20%7D%7Bh%7D%5Cright%29%7D%2B4h%20%3D%20%5Csqrt%7B%5Cfrac%7B1%2C280%7D%7Bh%7D%7D%2B4h%20)
Area of the four side faces of the aquarium is 4 times the length of the base times the height =
![4\times\sqrt{ \frac{20}{h} }\times h=\sqrt{16\left(\frac{20}{h}\right)h^2}=\sqrt{320h}](https://tex.z-dn.net/?f=4%5Ctimes%5Csqrt%7B%20%5Cfrac%7B20%7D%7Bh%7D%20%7D%5Ctimes%20h%3D%5Csqrt%7B16%5Cleft%28%5Cfrac%7B20%7D%7Bh%7D%5Cright%29h%5E2%7D%3D%5Csqrt%7B320h%7D)
Total area to be covered by grass is the base and the four side faces and is given by
![\frac{20}{h}+\sqrt{320h}](https://tex.z-dn.net/?f=%5Cfrac%7B20%7D%7Bh%7D%2B%5Csqrt%7B320h%7D)
Cost of the entire metal frame =
![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](https://tex.z-dn.net/?f=7%5Cleft%28%5Csqrt%7B%5Cfrac%7B1%2C280%7D%7Bh%7D%7D%2B4h%5Cright%29%3D%20%5Csqrt%7B49%5Cleft%28%5Cfrac%7B1%2C280%7D%7Bh%7D%5Cright%29%7D%2B7%284h%29%20%3D%20%5Csqrt%7B%5Cfrac%7B62%2C720%7D%7Bh%7D%7D%2B28h%20)
Cost of the entire grass =
![8\left(\frac{20}{h}+\sqrt{320h}\right)=\frac{160}{h}+\sqrt{64(320h)}=\frac{160}{h}+\sqrt{20,480h](https://tex.z-dn.net/?f=8%5Cleft%28%5Cfrac%7B20%7D%7Bh%7D%2B%5Csqrt%7B320h%7D%5Cright%29%3D%5Cfrac%7B160%7D%7Bh%7D%2B%5Csqrt%7B64%28320h%29%7D%3D%5Cfrac%7B160%7D%7Bh%7D%2B%5Csqrt%7B20%2C480h)
Therefore, total cost in terms of the height, h, is given by