Answer:
The dimensions should be length=width=height=74 cm.
Explanation:
We have an optimization with restriction problem.
We have to maximize the volume, subject to restriction in the sum of the length, width and height.
Let x be the length and width, that are equal, and z be the height.
The restriction can be expressed as:
![x+x+z\leq222\\\\2x+z\leq222](https://img.qammunity.org/2021/formulas/mathematics/college/waslc7l72hb9ce7gd8xyuq0phczcibeg23.png)
We can express z in function of x as:
![2x+z=222\\\\z=222-2x](https://img.qammunity.org/2021/formulas/mathematics/college/sa1u6s4gxxqt9p0uh1vxwaii61i6n3167m.png)
The volume, the function to be optimized, can be expressed as:
![V=x^2z=x^2(222-2x)=222x^2-2x^3](https://img.qammunity.org/2021/formulas/mathematics/college/mf98iquxmxz6u543xqjw3ztlbanpciy8y5.png)
To optimize, we derive and equal to zero.
![(dV)/(dx)=(d)/(dx)[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74](https://img.qammunity.org/2021/formulas/mathematics/college/lmtjd7hlp75gr0on92y7xbyln14hh9z53k.png)
We have the optimum length. We can now calculate the height z:
![z=222-2(74)=222-148=74](https://img.qammunity.org/2021/formulas/mathematics/college/kzoqj755khqew35rwbakaghgmmgltmkxau.png)