Chapter 14.5, Problem 25E

### 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 (in centimeters) of the box with square base, open top, and volume 500,000 cc 3 that requires the least materials.

To determine

To calculate: The dimensions (in centimeters) of the box with the square base, open top, and volume 500,000 cm3 which requires the least materials.

Explanation

Given Information:

The volume of the box with square base and open top is 500,000Â cm3.

Formula used:

Lagrange Multipliers Method:

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

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

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

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

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

For a function f(x,y), the partial derivative of f(x,y) 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(x,y) with respect to y is denoted by fy.

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).

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

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

Calculation:

Recall that, the surface area of a box with open top and length l, width w, and height h is 2lh+2wh+lw.

Let x be the length and width of the side of the square base as it is a square base so both length and width are equal, and y be the height of the box.

Thus, the surface area of a box with open top and length x, width x, and height h is 2xy+2xy+x2=4xy+x2.

Thus, minimize the function f(x,y)=4xy+x2.

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

Since, x is the length and width of the side of the square base and y be the height of the box.

Thus, the volume of the box is given by xâ‹…xâ‹…y=x2y. But the volume of the box with square base and open top is 500,000Â cm3. Thus, x2y=500,000.

The provided constraint is x2y=500,000.

According to the Lagrange multipliers method,

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

Here, f(x,y)=4xy+x2 and g(x,y)=x2yâˆ’500,000.

Substitute 4xy+x2 for f(x,y) and x2yâˆ’500,000 for g(x,y) in F(x,y,Î»)=f(x,y)+Î»g(x,y):

F(x,y,Î»)=4xy+x2+Î»(x2yâˆ’500,000)

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

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

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