Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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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: Graphical, Numerical, Algebraic

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