   Chapter 14.4, Problem 28E 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

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

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