Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 5.4, Problem 19E

To determine

Write a formula for the volume of the box.

Expert Solution

Answer to Problem 19E

The volume of the box is 8x3−168x2+864x

Explanation of Solution

Calculation:

The height of the box is 2x and the dimensions of the base are 18−2x and 24−2x .

Volume of box is

V=lwh =(18−2x)(24−2x)(2x) =(432−48x−36x+4x2)(2x) =8x3−168x2+864x

Thus, the volume of the box is 8x3−168x2+864x

To determine

Find the domain of V .

Expert Solution

Answer to Problem 19E

Explanation of Solution

Calculation:

volume from part (a)

V=2x(24−2x)(18−2x)

Calculus: Graphical, Numerical, Algebraic, Chapter 5.4, Problem 19E , additional homework tip 1

Domain is 0<x<9

To determine

Find the maximum volume and value of x .

Expert Solution

Answer to Problem 19E

Explanation of Solution

Calculation:

Examine the graph; the maximum occurs at about x=3.4 with volume slightly over 1300

Calculus: Graphical, Numerical, Algebraic, Chapter 5.4, Problem 19E , additional homework tip 2

Thus, the value is x≈3.4inches, volume a bit over1300in3

To determine

Support the result in part (c) graphically.

Expert Solution

Answer to Problem 19E

Explanation of Solution

Calculation:

Multiply it out to avoid product rule before find V′

V=(18−2x)(24−2x)(2x) =4x(12−x)(18−2x) =8(x3−21x2+108x)V′=8(3x2−14x+36) =24(x2−14x+36)Find whereV′=00=(x2−14x+36)x=−b±b2−4ac2a =−(−14)±(−14)2−4(1)(36)2(1) =14±4⋅132(1) =14±2132(1) =7+13≈10.61 =7−13≈3.39

Thus, the values are 7−13 and volume ≈1309.95in3

To determine

Find value of x that yields a volume of 1120in3 .

Expert Solution

Answer to Problem 19E

Explanation of Solution

Calculation:

Multiply it out to avoid product rule before find V′

V=(36−4x)∗(24−2x)∗x1120=2(18−2x)∗(12−x)∗x280=216x−24x2−18x2+2x3280=2x3−42x2+216xx3−21x2+108x−140=0 cancel 2 from bothsides(x−14)(x−5)(x−2)=0 x=2.5

To determine

Write a paragraph describe the issues that arise in part (b)

Expert Solution

Answer to Problem 19E

Explanation of Solution

Calculation:

Must be careful to restrict the size of x so that one square on a corner does not overlap a square on another corner. If they did overlap then there would be miss sides for the box or nothing left.

Chapter 5.4, Problem 18E

Chapter 5.4, Problem 20E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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