Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 5, Problem 51RE

To determine

To write a formula V(x) for the volume of box.

Expert Solution

Answer to Problem 51RE

The required formula is V(x)=2x3−25x2+75x .

Explanation of Solution

Given information: A piece of cardboard measures 10 in. by 15 in. Two equal sides are removed from the corners of a 15 in. sides.

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 1

Calculation: The required formula is obtained as,

Refer to the diagram exercise 18 of the sction 4.4, but imagine the vertical side as 15” and the horizontal side as 10” .

The length of box be 15−2x .

The width of box be 10−2x2=5−x .

The height of box be x .

V(x)=x(15−2x)(5−x) =2x3−25x2+75x

Hence, the required formula is V(x)=2x3−25x2+75x .

To determine

To find the domain of V for the problem situation and graph V over this domain.

Expert Solution

Answer to Problem 51RE

The domain is: 0 < x< 5 and graph below,

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 2

Explanation of Solution

Given information: A piece of cardboard measures 10 in. by 15 in. Two equal sides are removed from the corners of a 15 in. sides.

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 3

Calculation: The required solution is obtained as,

x must be > 0 since you have to cut something. x must be less than the smaller of 15/2 = 7.5 and 10/2 = 5, so that you don’t cut away more material than you actually have.

Hence, the domain is: 0 < x< 5 and graph below,

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 4

To determine

To find the maximum volume and the value of x that gives it, by using a graphical method.

Expert Solution

Answer to Problem 51RE

The maximum is at approximately x=1.96 and V=66.02 .

Explanation of Solution

Given information: A piece of cardboard measures 10 in. by 15 in. Two equal sides are removed from the corners of a 15 in. sides.

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 5

Calculation: The required solution is obtained as,

From the graph, the maximum is at approximately x=1.96 and V=66.02 .

To determine

To confirm your result in part (C) analytically.

Expert Solution

Answer to Problem 51RE

The required solution is 1.96.

Explanation of Solution

Given information: A piece of cardboard measures 10 in. by 15 in. Two equal sides are removed from the corners of a 15 in. sides.

Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 51RE , additional homework tip 6

Calculation: The required solution is obtained as,

Find V’ and where it is = 0.

V'(x)=6x2−50x+75 0=6x2−50x+75 x=−b±b2−4ac2a =−(−50)±(−50)2−4(6)(75)2(6) =50±70012 =50±10712 =25±576 =25+576≈6.37 =25−576≈1.96

Hence the required solution is 1.96.

Chapter 5, Problem 50RE

Chapter 5, Problem 52RE

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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