Chapter 14.4, Problem 27E

### Mathematical Applications for the ...

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

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 an open-top box of length x, width y, and height 500 , 000 / ( x y ) (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 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.

Explanation

Given Information:

The dimensions of an open-top box are given by length x, width y, and height 500,000/(xy) (in inches).

Formula used:

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

(1) Find the partial derivatives âˆ‚zâˆ‚x and âˆ‚zâˆ‚y.

(2) Find the critical points, that is, the point(s) that satisfy âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)âˆ’(zxy)2=âˆ‚2zâˆ‚x2â‹…âˆ‚2zâˆ‚y2âˆ’(âˆ‚2zâˆ‚xâˆ‚y)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=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

(2) When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then fâ€²(x)=v(x)â‹…uâ€²(x)âˆ’u(x)â‹…vâ€²(x)(v(x))2.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

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

Calculation:

Consider the problem, the dimensions of an open-top box are given by length x, width y, and height 500,000/(xy) (in inches).

Use the formula for the surface area of a box with open top,

A=2lh+2wh+lw

Substitute x for l, y for w, and 500,000/(xy) for h in A=2lh+2wh+lw.

A=2x(500,000xy)+2y(500,000xy)+xy=1,000,000y+1,000,000x+xy

Thus, the area function is A(x,y)=1,000,000y+1,000,000x+xy.

Use the power of x rule for derivatives, the constant function rule, the quotient rule, and the coefficient rule,

Thus,

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

âˆ‚Aâˆ‚x=âˆ‚âˆ‚x(1,000,000y+1,000,000x+xy)=âˆ’1,000,000x2+y

Equate the first derivative of function A(x,y)=1,000,000y+1,000,000x+xy with respect to x to zero.

âˆ’1,000,000x2+y=0y=1,000,000x2

The first partial derivative of function A(x,y)=1,000,000y+1,000,000x+xy with respect to y by holding x constant.

âˆ‚Aâˆ‚y=âˆ‚âˆ‚y(1,000,000y+1,000,000x+xy)=âˆ’1,000,000y2+x

Equate the first derivative of function A(x,y)=1,000,000y+1,000,000x+xy with respect to y to zero.

âˆ’1,000,000y2+x=0x=1,000,000y2

Now, calculate value of x and y

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