Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Chapter 2, Problem 27P

Area of a Box An open box with a square base is to have a volume of 12 ft3.

  1. (a) Find a function that models the surface area of the box.
  2. (b) Find the box dimensions that minimize the amount of material used.

(a)

Expert Solution
Check Mark
To determine

To find: The function that models the surface area of a open box with square base whose volume is 12 ft3 .

Answer to Problem 27P

The function that models the surface area of given box is S(x)=x2+48x .

Explanation of Solution

Let the dimension of square base be x and height h

Surface area (S) of a open box with square base is,

S=Area of four sides +Area of bottom=4(x⋅h)+(x2) (1)

Volume of the box is,

V=(Area of base )(height of box)=(x2)(h)=x2h

Substitute 12 for V in above equation.

12=x2h

Divide both sides of above equation by x2 .

12x2=h

Summarize all the information as shown in the table below.

In Words In Algebra
Surface Area 4(x⋅h)+(x2)
Length of side of square base. x
Height of the box 12x2

Use the information in the table and model the function.

S=4(x⋅h)+(x2)S(x)=4(x⋅(12x2))+(x2)=48x+x2

Thus, the function that models the surface area of given box is S(x)=x2+48x .

(b)

Expert Solution
Check Mark
To determine

To find: The dimensions of the box which minimize the material used to make the box.

Answer to Problem 27P

The length of square base of box is approximately 2.88 ft and height is 1.44 ft .

Explanation of Solution

The function as calculated in part (a) is,

S(x)=x2+48x

Sketch the graph of above function as shown below.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 2, Problem 27P

Figure (1)

Observe from the graph shown in Figure (1) that it attains minimum value at x=2.88

Length of square base is 2.88 for minimum surface area.

Height of box as calculated in terms of length of base in part (a) is,

h=12x2

Substitute 2.88 for x in above equation and solve for h.

h=12(2.88)2=128.29=1.44

Thus, the length of square base of box is approximately 2.88 ft and height is 1.44 ft .

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Chapter 2, Problem 28P

Chapter 2 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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Publisher:Cengage Learning
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