Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Chapter 5.4, Problem 64E

To determine

To find a formula for the volume V of the cone.

Expert Solution

Answer to Problem 64E

Thus formula for the volume V of the cone is V(x)=13π(2πa−x2π)2(a2−(2πa−x2π)2).

Explanation of Solution

Given r=2πa−x2π.

Calculation:

It is given in the question that r=2πa−x2π,h=a2−r2=a2−(2πa−x2π)2.

Let the volume of the cone is,

V=13π(r2)(h) =13π(2πa−x2π)2(a2−(2πa−x2π)2)V(x)=13π(2πa−x2π)2(a2−(2πa−x2π)2)

Thus formula for the volume V of the cone is V(x)=13π(2πa−x2π)2(a2−(2πa−x2π)2).

To determine

To find r and h in the cone of maximum value for a=4,5,6,8.

Expert Solution

Answer to Problem 64E

Hence r and h values are

a=4:r=463,h=433a=5:r=563,h=533a=6:r=663=26,h=633=23a=8:r=863,h=833

Explanation of Solution

Given r=2πa−x2π.

Calculation:

For any function f(x) the condition of minimum and maximum is obtained by find f′(x)=0 ,

Critical points and end points

V=13π(r2)(a2−r2)dVdr=π3(a2−r2)(2r)+π3(r2)(−2r)(2a2−r2)(−2a) =π3r(2a2−3r2)a2−r2

Critical points occur at

V′=0,0<r<aπ3r(2a2−3r2)a2−r24=02a2−3r2=0r=23a=a63

It has critical point occur at r=a63.

For,

r=a63h=(a2−r2) =400−200 =200≈14.14r=a33when,a=4:r=463,h=433a=5:r=563,h=533a=6:r=663=26,h=633=23a=8:r=863,h=833

Thus the r and h values are

a=4:r=463,h=433a=5:r=563,h=533a=6:r=663=26,h=633=23a=8:r=863,h=833

To determine

To find a simple relationship between r and h .

Expert Solution

Answer to Problem 64E

The value is rh=2 .

Explanation of Solution

Given r=2πa−x2π.

Calculation:

r=a63,h=a33rh=2

Thus the value is rh=2 .

Chapter 5.4, Problem 63E

Chapter 5.4, Problem 65E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

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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