Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Differential Equations

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Question

Chapter 5.5, Problem 46E

To determine

To calculate the accuracy you should maintain in measure the interior diameter of the cylinder.

Expert Solution

Answer to Problem 46E

You should measure r with an error dr that is no more than 0.5% of the true value.

Explanation of Solution

Given information:

Height of the cylinder is 10 m.

Be sure of calculating the volume of cylinder to within 1% of its true value.

Calculation :

Since percentage error allowed for the radius is same as the percentage error allowed for diameter.

Consider the interior diameter of the cylinder be r .

You want any accuracy in your measurement to be small enough to make the corresponding increment ΔV in the volume satisfy the inequality.

|ΔV|≤1100V=πr2h100 =πr2⋅10100 [h=10]

Replace ΔV in this inequality by its approximation

dV=(dVdr)dr =20πrdr [h=10]

This gives

|20πrdr|≤πr2⋅10100or,|dr|≤120πr⋅πr2⋅10100 =12⋅r100 =0.005r

You should measure r with an error dr that is no more than 0.5% of the true value.

To determine

To calculate the accuracy you should maintain in measure the exterior diameter of the cylinder.

Expert Solution

Answer to Problem 46E

You should measure r with an error dr that is no more than 5% of the true value.

Explanation of Solution

Given information:

Height of the cylinder is 10 m.

Be sure of calculating the surface area of side wall of cylinder to within 5% of its true value.

Calculation :

Since percentage error allowed for the radius is same as the percentage error allowed for diameter.

Consider the exterior diameter of the cylinder be r .

You want any accuracy in your measurement to be small enough to make the corresponding increment ΔS in the volume satisfy the inequality.

|ΔS|≤5100S=10πrh100 =10πr⋅10100 [h=10]

Replace ΔV in this inequality by its approximation

dS=(dSdr)dr =20πdr [h=10]

This gives

|20πdr|≤10πr⋅10100or,|dr|≤120π⋅10πr⋅10100 =r20 =0.05r

You should measure r with an error dr that is no more than 5% of the true value.

Chapter 5.5, Problem 45E

Chapter 5.5, Problem 47E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

Ch. 5.1 - Prob. 11QR Ch. 5.1 - Prob. 12QR Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - 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