   Chapter 10.4, Problem 30E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Minimum material(a) A box with an open top and a square base is to be constructed to contain 4000 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.(b) A box with an open top and a square base is to be constructed to contain k cubic inches. Show that the minimum amount of material is used to construct the box when each side of the base is x =   ( 2 k ) l / 3 and the height is y =   ( k / 4 ) 1 / 3

(a)

To determine

To calculate: The dimensions to minimize the material to construct the box.

Explanation

Given Information:

Volume of the box is 4000 cubic inches with an open top and square base.

Formula used:

To find the minimum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is more than zero, then substituting the value of the independent variable will give the minimum value of the equation.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function f(x)=xn, the derivative is

ddx[xn]=nxn1

Calculation:

Consider the provided statement,

Volume of the box is 4000 cubic inches with an open top and square base.

Assume side of the square base is x and other side of the box other than x is y

The surface area of the square base box, which has an open top is:

SA=x2+4xy

Use the equation of volume x2y=4000 and calculate y in terms of x,

y=4000x2

Substitute y=4000x2 in SA=x2+4xy,

SA=x2+4x(4000x2)SA=x2+16000x

To find the stationary value for SA=x2+16000x, differentiate the equation with respect to the independent variable x to get

SA'=2x16000x2

Put SA'=0 to get x,

2x=16000x2x=1600023x=±

(b)

To determine

To prove: The dimensions are x=(2k)13 for side and y=(k4)13 for height to minimize the material used if volume of the box is k cubic inches with an open top and square base.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Exercises 1-22, evaluate the given expression. C(5,0)

Finite Mathematics for the Managerial, Life, and Social Sciences

#### The slope of the tangent line to r = cos θ at is:

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

#### Define fraud and explain the safeguards that exist to prevent it.

Research Methods for the Behavioral Sciences (MindTap Course List) 