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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Volume Find the dimensions of the rectangular package of maximum volume subject to the constraint that the sum of the length and the girth cannot exceed 108 inches (see figure). (Hint: Maximize V = xyz subject to the constraint x + 2y + 2z = 108.)

Chapter 7.6, Problem 30E, Volume Find the dimensions of the rectangular package of maximum volume subject to the constraint

To determine

To calculate: The dimensions of the rectangular passage that maximize the volume V=xyz and x+2y+2z=108.

Explanation

Given Information:

The rectangular passage that maximize the volume V=xyz and x+2y+2z=108.

Formula used:

The Lagrange Multiplier,

f(x,y,z)=λg(x,y,z)

Where, f(x,y,z)andλg(x,y,z) are the function and the constraint and λ is the Lagrange Multiplier.

Step 1: Write the function and the constraint values.

Step 2: Partially differentiate 'f'and'g' with respect to x,yandz.

Step 3: Using the primary formula, find the value of λ in each case.

Step 4: Find the relation between the dimensions x,yandz.

Step 5: Substitute them half into the constraint equation.

Step 6: Substitute the value of the found dimension in the constraint equation to obtain the other dimension.

Calculation:

Given that f(x,y,z) as V=xyz and g(x,y,z) as x+2y+2z=108

Now partially differentiating f(x,y,z) with respect to x,yandz,

dfdx=yzdfdy=xzdfdz=xy

Similarly, partially differentiating g(x,y,z) with respect to x,yandz,

dgdx=1

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