   Chapter 7.6, Problem 29E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Volume A rectangular box is resting on the xy-plane with one vertex at the origin. The opposite vertex lies in the plane 2 x + 3 y + 5 z = 90. Find the dimensions that maximize the volume. (Hint: Maximize V = xyz subject to the constraint 2 x + 3 y + 5 z − 90 = 0. )

To determine

To calculate: The dimensions of the rectangular box that maximize the volume V=xyz and 2x+3y+5z=90

Explanation

Given Information:

The rectangular box that maximize the volume V=xyz and 2x+3y+5z=90,.

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 2x+3y+5z=90

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=2dgdy=3dgdz=5

Consider the primary equation and substitute the values of f and g,

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