(a) Answer The volume of the box is 2 x 3 − 25 x 2 + 75 x . Explanation Given: The given diagram is shown in Figure 1 Figure 1 Calculation: Consider the dimension of the base is ( 10 − 2 x ) in by ( 15 − 2 x 2 ) in . Then, the volume of the box is, V = x ( 15 − 2 x 2 ) ( 10 − 2 x ) = x ( 75 − 25 x + 2 x 2 ) = 2 x 3 − 25 x 2 + 75 x (b) To determine To find: The domain of V for the problem situation and the graph V over the domain. Answer The domain is ( 0 , 5 ) and the graph is shown in Figure 2 Explanation Calculation: Consider the range required to make the box is, x > 0 2 x < 10 2 x < 15 Consider the interval is ( 0 , 5 ) . The graph is shown in Figure 2 Figure 2 (c) To determine To find: The maximum value and the value of x that gives it. Answer The maximum value is at x = 1.96 in that is 66.02 in 3 . Explanation Calculation: Consider that the maximum value is obtained at x = 1.96 in . Then, V = 2 ( 1.96 ) 3 − 25 ( 1.96 ) 2 + 75 ( 1.96 ) = 66.02 in 2 (d) To determine To find: The confirmation for the result of part (c). Answer The maximum value occur at x = 1.96 in . Explanation Calculation: Consider the maximum value occur at, V ′ = 0 6 x 2 − 50 x + 75 = 0 x = 1.96 , 6.37 Thus, it is clear that the maximum value occur at x = 1.96 in .

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

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5.4, Problem 18E

(a)

To determine

To find: The formula for the V(x) for the volume of the box.

(a)

Expert Solution

Answer to Problem 18E

The volume of the box is 2x3−25x2+75x .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.4, Problem 18E , additional homework tip 1

Figure 1

Calculation:

Consider the dimension of the base is (10−2x) in by (15−2x2) in .

Then, the volume of the box is,

V=x(15−2x2)(10−2x)=x(75−25x+2x2)=2x3−25x2+75x

(b)

To determine

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

(b)

Expert Solution

Answer to Problem 18E

The domain is (0,5) and the graph is shown in Figure 2

Explanation of Solution

Calculation:

Consider the range required to make the box is,

x>02x<102x<15

Consider the interval is (0,5) .

The graph is shown in Figure 2

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.4, Problem 18E , additional homework tip 2

Figure 2

(c)

To determine

To find: The maximum value and the value of x that gives it.

(c)

Expert Solution

Answer to Problem 18E

The maximum value is at x=1.96 in that is 66.02 in3 .

Explanation of Solution

Calculation:

Consider that the maximum value is obtained at x=1.96 in .

Then,

V=2(1.96)3−25(1.96)2+75(1.96)=66.02 in2

(d)

To determine

To find: The confirmation for the result of part (c).

(d)

Expert Solution

Answer to Problem 18E

The maximum value occur at x=1.96 in .

Explanation of Solution

Calculation:

Consider the maximum value occur at,

V′=06x2−50x+75=0x=1.96,6.37

Thus, it is clear that the maximum value occur at x=1.96 in .

Chapter 5.4, Problem 17E

Chapter 5.4, Problem 19E

Chapter 5 Solutions

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

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