To determine
To estimate: The maximum possible error, relative error and percentage error when computing the volume of the cube.
Answer
The maximum possible error is dV=270 cm3, the relative error is 0.01 and the percentage error is 1%.
Explanation
Given:
The edge of the cube is 30 cm and possible error in measurement is 0.1 cm.
Calculation:
Let x be the length of the side of the cube.
The volume of the cube is V=x3.
Consider the function f(x)=x3
The differential is dV=f′(x)dx (1)
Derivative of the function f(x)=x3 is f′(x)=3x2.
Substitute f′(x)=3x2 in equation (1),
dV=3x2dx
Substitute x=30 and dx=0.1,
dV=3(30)2(0.1)=3(900)(0.1)=2700(0.1)=270
Thus, the maximum possible error in computing the volume of the cube is dV=270 c.m3
Note that, the relative error is ΔVV and ΔV is approximately equal to dV.
ΔVV≈dVV=3x2dxx3=3dxx
Substitute x=30 and dx=0.1,
ΔVV≈3(0.1)30=0.01
Therefore, the relative error is 0.01.
Note that, the percentage error is the product of relative error and 100%.
The percentage error is computed as follows,
Percentage error=0.01×100%=1%
Therefore, the percentage error is 1%.
To determine
To estimate: The maximum possible error, relative error and percentage error when computing the surface area of the cube.
Answer
The maximum possible error is dS=36 cm2, the relative error is 0.006¯.and the percentage error is 0.6¯ %.
Explanation
Given:
The edge of the cube is 30 cm and possible error in measurement of 0.1 cm.
Calculation:
The surface of the cube is S=6x2.
Consider the function f(x)=6x2
Derivative of the function f(x)=6x2 is f′(x)=12x.
Substitute f′(x)=12x in the differential dS=f′(x)dx,
dS=12xdx=12(30)(0.1) (x=30)=360(0.1)=36
Therefore, the maximum possible error in computing the surface area of the cube is dS=36 cm2
The relative error is computed by dividing the change of the surface area of the cube (ΔS) by the surface area of the cube S.
That is, the relative error is ΔSS.
Note that, the value ΔS is approximately equal to dS.
ΔSS≈dSS=12xdx6x2=2dxx
Substitute x=30 and dx=0.1,
ΔSS≈2(0.1)30=0.230=0.006¯
Therefore, the relative error is 0.006¯.
Note that, the percentage error is the product of relative error and 100%.
The percentage error is computed as follows,
Percentage error=0.006¯×100%=0.6¯ %
Therefore, the percentage error is 0.6¯ %.