   Chapter 4.7, Problem 30E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Maximum Volume Rework Exercise 29 for a cylindrical package. (The cross section is circular.)

To determine

To calculate: The dimensions of the cylindrical box that gives the maximum volume.

Explanation

Given:

The cylindrical package has a cross sectional perimeter of 108 inches.

Formula used:

Let the volume of the box be denoted by V and r denote the radius of the base and h denote the height of the box.

V=πr2h

For a function f that is twice differentiable on an open interval I, if f'(c)=0 for some c, then,

If f''(c)>0 the function f has relative minima at c if f''(c)<0 the function f has relative maxima at c.

Calculation:

Let the volume of the box be denoted by V and x denote the radius of the base and y denote the height of the box.

V=πx2y

It has also been provided that the package has a cross sectional perimeter of 108 inches. This implies:

2πx+y=108y=1082πx

Substitute this back in the volume to obtain:

V=πx2(1082πx)=108πx22π2x3

Differentiate the provided function:

V'(x)=216πx6π2x

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