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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm.

(a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error?

(b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error?

(a)

To determine

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

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

(b)

To determine

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

and obtain the relative error

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