#### To determine

**To estimate:** The maximum error in the surface area of the sphere by using differentials and obtain the relative error.

#### Answer

The maximum error in the surface of area dS≈27 cm3 and the relative error is approximately 0.012 or 1.2%.

#### Explanation

**Given:**

The circumference of the sphere is 84 cm with possible error is 0.5 cm.

**Calculation:**

The circumference of the sphere with radius *r* is, C=2πr.

r=C2π

The surface area of the sphere with radius *r* is, S=4πr2.

Substitute r=C2π in S=4πr2,

S=4π(C2π)2=4π(C24π2)=C2π

The differential is dS=f′(C)dC,

The derivative of the function f(C) is computed as follows,

f′(C)=ddC(C2π)=1π(2C)=2Cπ

Substitute f′(C)=2Cπ in dS=f′(C)dC,

dS=2CπdC

Substitute the value C=84 and dC=0.5,

dS=2(84)π(0.5)=84π≈27 cm3

Therefore, the maximum error in the surface of area dS≈27 cm3.

The relative error is computed by dividing the error (ΔS) by the total surface of area *S* . That is, the relative error is ΔSS.

ΔSS≈dSS=2CπdCC2π=2dCC

Substitute C=84 and dC=0.5,

ΔSS≈2(0.5)84=184≈0.012

Therefore, the relative error is approximately 0.012 or 1.2%.

#### To determine

**To estimate:** The maximum error in the volume of the sphere by using differentials

and obtain the relative error

#### Answer

The maximum error in the surface of area of the sphere is dV≈179 cm3 and the relative error is approximately 1.8%.

#### Explanation

**Calculation:**

The volume of the sphere with radius *r* is V=4πr33 .

Substitute r=C2π in V=4πr33,

V=4π(C2π)33=4πC38π33=C36π2

The differential is dV=f′(C)dC,

The derivative of the function f(C) is computed as follows,

f′(C)=ddC(C36π2)=16π2(C3)=16π2(3C2)=C22π2

Substitute f′(C)=C22π2 in dV=f′(C)dC,

dV=(C22π2)dC

Substitute the value C=84 and dC=0.5,

dV=(84)22π2(0.5)=(70562π2)(0.5)=1764π2≈179 cm3

Therefore, the maximum error in the surface of area of the sphere is dV≈179 cm3.

The relative error is computed as follows,

ΔVV≈dVV=1764π2C36π2=1764C3(6)=10584C3

Substitute C=84,

ΔVV≈10584(84)3=10584592704≈0.018≈1.8%

Therefore, the relative error is approximately 1.8%.