Answer:
Maximum possible error volume
![1080cm^3](https://tex.z-dn.net/?f=1080cm%5E3)
Maximum possible error surface
![144cm^2](https://tex.z-dn.net/?f=144cm%5E2)
Relative error volume
0.04
Relative error surface
0.0266
% error volume
4%
% error surface
2.66%
Step-by-step explanation:
Let's call x the length of the edge of the cube, V its volume and S its surface.
We have x= 30 cm with a possible error of 0.4 cm
On the other hand,
![V=x^3](https://tex.z-dn.net/?f=V%3Dx%5E3)
and
![S=6x^2](https://tex.z-dn.net/?f=S%3D6x%5E2)
Taking differentials
![\frac{dV}{dx}=3x^2](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdx%7D%3D3x%5E2)
and
![\frac{dS}{dx}=12x](https://tex.z-dn.net/?f=%5Cfrac%7BdS%7D%7Bdx%7D%3D12x)
So, the error when estimating the volume and surface, dV, dS are
![dV=3x^2dx](https://tex.z-dn.net/?f=dV%3D3x%5E2dx)
and
![dS = 12xdx](https://tex.z-dn.net/?f=dS%20%3D%2012xdx)
Where x = 30cm and dx=0.4cm
Replacing
![dV=3(30)^20.4=3(900)0.4=1080cm^3](https://tex.z-dn.net/?f=dV%3D3%2830%29%5E20.4%3D3%28900%290.4%3D1080cm%5E3)
![dS=12(30)0.4=144cm^2](https://tex.z-dn.net/?f=dS%3D12%2830%290.4%3D144cm%5E2)
The relatives errors are
![\frac{dV}{V}=\frac{3x^2dx}{x^3}=\frac{3dx}{x}=\frac{3(0.4)}{30}=\frac{1.2}{30}=0.04](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7BV%7D%3D%5Cfrac%7B3x%5E2dx%7D%7Bx%5E3%7D%3D%5Cfrac%7B3dx%7D%7Bx%7D%3D%5Cfrac%7B3%280.4%29%7D%7B30%7D%3D%5Cfrac%7B1.2%7D%7B30%7D%3D0.04)
![\frac{dS}{S}=\frac{12xdx}{6x^2}=\frac{2dx}{x}=\frac{2(0.4)}{30}=\frac{0.8}{30}=0.0266](https://tex.z-dn.net/?f=%5Cfrac%7BdS%7D%7BS%7D%3D%5Cfrac%7B12xdx%7D%7B6x%5E2%7D%3D%5Cfrac%7B2dx%7D%7Bx%7D%3D%5Cfrac%7B2%280.4%29%7D%7B30%7D%3D%5Cfrac%7B0.8%7D%7B30%7D%3D0.0266)
The percentage errors are relative errors times 100%
![\frac{dV}{V}100\%=(0.04)100\%=4\%](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7BV%7D100%5C%25%3D%280.04%29100%5C%25%3D4%5C%25)
![\frac{dS}{S}100\%=(0.0266)100\%=2.66\%](https://tex.z-dn.net/?f=%5Cfrac%7BdS%7D%7BS%7D100%5C%25%3D%280.0266%29100%5C%25%3D2.66%5C%25)