#### 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¯ %.