   Chapter 4.7, Problem 17E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Do Exercise 16 assuming the container has a lid that is made from the same material as the sides.

To determine

To find: The cost of the cheapest container has a lid that is made from the same material as the sides.

Explanation

Let the length of the container be denoted by x m.

Width is denoted by y m.

Height is denoted by z m.

Volume= x×y×z=10m3 .

Cost (C) = 10(x×y)+6(x×y)+6(2×x×z+2×y×z)

Given as follows x=2y

Substitute the value of x=2y in volume,

2y2z=10m3

z=5y2m

Substitute the value of z=5y2 , x=2y in cost,

C=16×x×y+12×x×z+12×y×z=16×2×y×y+12×2×y×5y2+12×y×5y2=32y2+6(20y+10y)

Differentiate with respect to y,

dCdy=64y180y2

For critical points, dCdy=0 .

64y180y2 =0

64y3=180y=180643mx=2180643mz=5(18064)23m

Differentiate dCdy with respect to y,

d2Cdy2=64180×(2)×y3=64+360×y3

Substitute the value of y=180643m in d2Ady2 then,

d2Cdy2=64+360×y3=64+360×(180643)3

d2Cdy2 is positive

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 