Chapter 14.5, Problem 24E

### 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 dimensions x, y, and z (in inches) of the rectangular box with the largest volume that satisfies 3 x   + 4 y +   12 z   =   12

To determine

To calculate: The dimensions x, y and z (in inches) of the rectangular box with the largest volume that satisfies 3x+4y+12z=12.

Explanation

Given Information:

The provided constraint is 3x+4y+12z=12.

Formula used:

Lagrange Multipliers Method:

According to the Lagrange multipliers method to obtain maxima or minima for a function w=f(x,y,z) subject to the constraint g(x,y,z)=0,

Step 1: Find the critical values of f(x,y,z) using the new variable Î» to form the objective function F(x,y,z,Î»)=f(x,y,z)+Î»g(x,y,z).

Step 2: The critical points of f(x,y,z) are the critical values of F(x,y,z,Î») which satisfies g(x,y,z)=0.

Step 3: The critical points of F(x,y,z,Î») are the points that satisfy:

âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, âˆ‚Fâˆ‚z=0 and âˆ‚Fâˆ‚Î»=0, that is, the points which make all the partial derivatives equal to zero.

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

And, similarly the partial derivatives of function f(x,y,z) with respect to xÂ andÂ z can be calculated as above.

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

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).

Calculation:

Recall that, the volume of a rectangular box with length l, width w, and height h is lwh.

Since, the dimensions provided in the problem are x, y and z.

Therefore, the volume of the rectangular box is xyz.

Thus, minimize the function f(x,y,z)=xyz.

Also, the provided constraint is 3x+4y+12z=12.

According to the Lagrange multipliers method,

The objective function is F(x,y,z,Î»)=f(x,y,z)+Î»g(x,y,z).

Thus, f(x,y,z)=xyz and g(x,y,z)=3x+4y+12zâˆ’12.

Substitute xyz for f(x,y,z) and 3x+4y+12zâˆ’12 for g(x,y,z) in F(x,y,z,Î»)=f(x,y,z)+Î»g(x,y,z).

F(x,y,z,Î»)=xyz+Î»(3x+4y+12zâˆ’12)

Since, the critical points of F(x,y,z,Î») are the points that satisfy:

âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, âˆ‚Fâˆ‚z=0 and âˆ‚Fâˆ‚Î»=0.

Recall that, for a function f(x,y,z), the partial derivative of f(x,y) with respect to y is calculated by taking the derivative of f(x,y,z) with respect to y and keeping the other variables x and z constant

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