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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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?

To determine

(a)

To estimate:

The maximum possible error and the relative error in the calculated surface area, using differentials.

Explanation

1) Formula:

i. Surface area of a sphere is,

S=4πr2, where r be the radius of a sphere

ii. The circumference of a sphere is,

C=2πr, where r is the radius of a sphere

iii. From the above expression,

r= C2π

iv. Relative error is calculated by dividing the maximum possible error by the total surface area:

ΔSS dSS

2) Given: C = 84 cm and ΔC = 0.5 cm

3) Calculation:

To find maximum possible error in the calculated surface area express S as a function of C. We are given that the circumference of the sphere is 84 cm with possible error of 0.5cm.

Substitute

r= C2π

in S=4πr2

S=4π(C2π)2=  4πC24π2=C2π

S= C2π

Now differentiate

S= C2π

with respect to C.

dSdC=2Cπ

Multiply both sides by dC

dS=2Cπ dC

ΔS 2CπΔC

Substitute C = 84 cm and ΔC=0.5 cm in

ΔS 2CπΔC

ΔS 2CπΔC=  2(84)(0

To determine

b)

To estimate:

The maximum possible error and the relative error in the calculated volume using differentials.

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