BuyFindarrow_forward

Mathematical Applications for the ...

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

Solutions

Chapter
Section
BuyFindarrow_forward

Mathematical Applications for the ...

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

Manufacturing Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 27,000 cubic inches and the box requires the least amount of material to make. (Hint: First write the height in terms of x and y, as in Problem 27.)

Manufacturing Find the values for each of the dimensions of an open-top box of length x, width y, and height 500,000/(xy) (in inches) such that the box requires the least amount of material to make.

To determine

To calculate: The values of each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 27,000 cubic inches and the box requires the least amount of material to make.

Explanation

Given Information:

The dimensions of a closed-top box are given by length x, width y, and height z (in inches) and the volume of the box is 27,000 cubic inches.

Formula used:

To calculate relative maxima and minima of the function z=f(x,y),

(1) Find the partial derivatives zx and zy.

(2) Find the critical points, that is, the point(s) that satisfy zx=0 and zy=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)(zxy)2=2zx22zy2(2zxy)2.

(a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0.

(b) If D<0, then neither a relative maximum nor a relative minimum occurs.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by fx and with respect to y is denoted by fy.

For a function z(x,y), the second partial derivative,

(1) When both derivatives are taken with respect to x is zxx=2zx2=x(zx).

(2) When both derivatives are taken with respect to y is zyy=2zy2=y(zy).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy).

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=v(x)u(x)u(x)v(x)(v(x))2.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

The surface area of a box with closed top and length l, width w, and height h is A=2lh+2wh+2lw.

The volume of a box with length l, width w, and height h is lwh.

Calculation:

Consider the problem, the dimensions of a closed-top box are given by length x, width y, and height z (in inches) and the volume of the box is 27,000 cubic inches.

Use the formula for the surface area of a box with closed top and length l, width w, and height h.

A=2lh+2wh+2lw.

Substitute x for l, y for w, and z for h in A=2lh+2wh+2lw.

A=2xz+2yz+2xy

Use the formula for the volume of a box with length l, width w, and height h.

V=lwh

Substitute x for l, y for w, and z for h in V=lwh.

V=xyz

Since, the volume of the box is 27,000 cubic inches.

Substitute 27,000 for V.

27,000=xyzz=27,000xy

Substitute 27,000xy for z in A=2xz+2yz+2xy.

A=2x(27,000xy)+2y(27,000xy)+2xy=54,000y+54,000x+2xy

Thus, the area function is A(x,y)=54,000y+54,000x+2xy.

Differentiate A=f(x,y) with respect to x by holding y constant,

Ax=x(54,000y+54,000x+2xy)=54,000x2+2y

Equate the first derivative of function A(x,y)=54,000y+54,000x+2xy with respect to x to zero.

54,000x2+2y=02y=54,000x2y=27,000x2

The first partial derivative of function A(x,y)=54,000y+54,000x+2xy with respect to y by holding x constant.

Ay=y(54,000y+54,000x+2xy)=54,000y2+2x

Equate the first derivative of function A(x,y)=54,000y+54,000x+2xy with respect to y to zero

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

Chapter 14 Solutions

Show all chapter solutions add
Sect-14.1 P-10ESect-14.1 P-11ESect-14.1 P-12ESect-14.1 P-13ESect-14.1 P-14ESect-14.1 P-15ESect-14.1 P-16ESect-14.1 P-17ESect-14.1 P-18ESect-14.1 P-19ESect-14.1 P-20ESect-14.1 P-21ESect-14.1 P-22ESect-14.1 P-23ESect-14.1 P-24ESect-14.1 P-25ESect-14.1 P-27ESect-14.1 P-28ESect-14.1 P-29ESect-14.1 P-30ESect-14.1 P-31ESect-14.1 P-32ESect-14.1 P-33ESect-14.1 P-34ESect-14.1 P-35ESect-14.1 P-36ESect-14.1 P-37ESect-14.1 P-38ESect-14.2 P-1CPSect-14.2 P-2CPSect-14.2 P-3CPSect-14.2 P-4CPSect-14.2 P-5CPSect-14.2 P-1ESect-14.2 P-2ESect-14.2 P-3ESect-14.2 P-4ESect-14.2 P-5ESect-14.2 P-6ESect-14.2 P-7ESect-14.2 P-8ESect-14.2 P-9ESect-14.2 P-10ESect-14.2 P-11ESect-14.2 P-12ESect-14.2 P-13ESect-14.2 P-14ESect-14.2 P-15ESect-14.2 P-16ESect-14.2 P-17ESect-14.2 P-18ESect-14.2 P-19ESect-14.2 P-20ESect-14.2 P-21ESect-14.2 P-22ESect-14.2 P-23ESect-14.2 P-24ESect-14.2 P-25ESect-14.2 P-26ESect-14.2 P-27ESect-14.2 P-28ESect-14.2 P-29ESect-14.2 P-30ESect-14.2 P-31ESect-14.2 P-32ESect-14.2 P-33ESect-14.2 P-34ESect-14.2 P-35ESect-14.2 P-36ESect-14.2 P-37ESect-14.2 P-38ESect-14.2 P-39ESect-14.2 P-40ESect-14.2 P-41ESect-14.2 P-42ESect-14.2 P-43ESect-14.2 P-44ESect-14.2 P-45ESect-14.2 P-46ESect-14.2 P-47ESect-14.2 P-48ESect-14.2 P-49ESect-14.2 P-50ESect-14.2 P-51ESect-14.2 P-52ESect-14.2 P-53ESect-14.2 P-54ESect-14.2 P-55ESect-14.2 P-56ESect-14.3 P-1CPSect-14.3 P-2CPSect-14.3 P-3CPSect-14.3 P-1ESect-14.3 P-2ESect-14.3 P-3ESect-14.3 P-4ESect-14.3 P-5ESect-14.3 P-6ESect-14.3 P-7ESect-14.3 P-8ESect-14.3 P-9ESect-14.3 P-10ESect-14.3 P-11ESect-14.3 P-12ESect-14.3 P-13ESect-14.3 P-14ESect-14.3 P-15ESect-14.3 P-16ESect-14.3 P-17ESect-14.3 P-18ESect-14.3 P-19ESect-14.3 P-20ESect-14.3 P-21ESect-14.3 P-22ESect-14.3 P-23ESect-14.3 P-24ESect-14.3 P-25ESect-14.3 P-26ESect-14.3 P-27ESect-14.3 P-28ESect-14.3 P-29ESect-14.3 P-30ESect-14.4 P-1CPSect-14.4 P-2CPSect-14.4 P-3CPSect-14.4 P-4CPSect-14.4 P-1ESect-14.4 P-2ESect-14.4 P-3ESect-14.4 P-4ESect-14.4 P-5ESect-14.4 P-6ESect-14.4 P-7ESect-14.4 P-8ESect-14.4 P-9ESect-14.4 P-10ESect-14.4 P-11ESect-14.4 P-12ESect-14.4 P-13ESect-14.4 P-14ESect-14.4 P-15ESect-14.4 P-16ESect-14.4 P-17ESect-14.4 P-18ESect-14.4 P-19ESect-14.4 P-20ESect-14.4 P-21ESect-14.4 P-22ESect-14.4 P-23ESect-14.4 P-24ESect-14.4 P-25ESect-14.4 P-26ESect-14.4 P-27ESect-14.4 P-28ESect-14.4 P-29ESect-14.4 P-30ESect-14.4 P-31ESect-14.4 P-32ESect-14.4 P-34ESect-14.4 P-35ESect-14.4 P-36ESect-14.5 P-1CPSect-14.5 P-2CPSect-14.5 P-3CPSect-14.5 P-4CPSect-14.5 P-1ESect-14.5 P-2ESect-14.5 P-3ESect-14.5 P-4ESect-14.5 P-5ESect-14.5 P-6ESect-14.5 P-7ESect-14.5 P-8ESect-14.5 P-9ESect-14.5 P-10ESect-14.5 P-11ESect-14.5 P-12ESect-14.5 P-13ESect-14.5 P-14ESect-14.5 P-15ESect-14.5 P-16ESect-14.5 P-17ESect-14.5 P-18ESect-14.5 P-19ESect-14.5 P-20ESect-14.5 P-21ESect-14.5 P-22ESect-14.5 P-23ESect-14.5 P-24ESect-14.5 P-25ESect-14.5 P-26ECh-14 P-1RECh-14 P-2RECh-14 P-3RECh-14 P-4RECh-14 P-5RECh-14 P-6RECh-14 P-7RECh-14 P-8RECh-14 P-9RECh-14 P-10RECh-14 P-11RECh-14 P-12RECh-14 P-13RECh-14 P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-1TCh-14 P-2TCh-14 P-3TCh-14 P-4TCh-14 P-5TCh-14 P-6TCh-14 P-7TCh-14 P-8TCh-14 P-9TCh-14 P-10T

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

In Exercises 75-98, perform the indicated operations and/or simplify each expression. 86. (5x + 2)(3x 4)

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Exercises 43 to 46, list all subsets of the given set. Ă˜

Mathematical Excursions (MindTap Course List)

Write each number in decimal form: 26106

Elementary Technical Mathematics

In Exercises 1926, pivot the system about the circled element. [123033041|523]

Finite Mathematics for the Managerial, Life, and Social Sciences

The curve given by r(t) = 2i + tj + 2tk is a: line plane spiral circle

Study Guide for Stewart's Multivariable Calculus, 8th

. ∞ −∞ 1 −1

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