Chapter 7.5, Problem 47E

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Cost A manufacturer makes an open-top wooden crate having a volume of 18 cubic feet. Material costs are $0.20 per square foot for the base and$0.15 per square foot for the sides. Find the dimensions that minimize the cost of each crate. What is the minimum cost?

To determine

To calculate: The dimensions that minimize the cost of each crate and the minimum cost. If The volume of an open-top wooden crate is 18 cubic feet. the cost of material is $0.20 per square foot for the base and$0.15 per square foot for the side.

Explanation

Given Information:

The provided figure is,

Formula used:

Consider a package (rectangular) of the following dimensions,

Then the volume is V=lbh.

Also, if there is a function f in a variable x, it’s critical point can be found by taking a derivative and setting it equal to zero,

fx=0

Step1: 2fx2>0 for the above critical point then f attains a minimum.

Step2: 2fx2<0 for the above critical point then f attains a maximum.

Calculation:

Let the length of the rectangular package be x, width be y and height be z.

Since, the volume of an open-top wooden crate is 18 cubic feet. so,

xyz=18

It can be written as z=18xy.

The cost of each crate is

C=0.2xy+0.15(2)xz+0.15(2)yz=0.2xy+0.3xz+0.3yz=0.2xy+0.3x(18xy)+0.3y(18xy)=0.2xy+5.4y+5.4x

Now find partial derivative of V with respect to y and z,

Cx=0.2y5

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