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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
Publisher: Brooks Cole
ISBN: 9781285464640

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BuyFindarrow_forward

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
Publisher: Brooks Cole
ISBN: 9781285464640
Chapter 8.5, Problem 26E
Textbook Problem
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Minimizing Construction Costs An open rectangular box is to be constructed from material that costs $3/ft2 for the bottom and $1/ft2 for its sides. Find the dimensions of the box of greatest volume that can be constructed for $36.

To determine

To find: The dimension of a rectangular package having the greatest possible volume.

Explanation of Solution

Given:

Volume of an open rectangular box is $36 constructed from a material costs $3/ft2 and $1/ft2 for the bottom and on its sides respectively.

Formula used:

Lagrange Multipliers:

To obtain the relative extremum of the function f(x,y,z) subject to the constraint g(x,y,z)=0 (assuming that these extreme values exist),

(1) Form an auxiliary function F(x,y,z,λ)=f(x,y,z)+λg(x,y,z) called the Lagrangian function where the variable λ is called the Lagrange multiplier.

(2) Solve the system that consists of the equations Fx=0 , Fy=0 , Fz=0 and Fλ=0 for all values of x, y, z and λ .

(3) The solutions found in step (2) are candidates for the extremum of f.

Calculation:

Suppose the side, top and bottom of the rectangular box is x, y and z respectively.

From given the open rectangular box has two sides of side and bottom and only one side of the top.

Then, the perimeter of the box is x+2(y+z) .

And the cost for box $3/ft2 on bottom and $1/ft2 on sides are multiplied for respective dimension on the boundary that is, x+2(y(3)+z(1)) .

Thus the cost associated with the perimeter of the open rectangular box is x+2(3y+z) .

The Volume of a rectangular box is $36 .

xyz=36xyz36=0

Obtain the maximum of f(x,y,z)=x+2(3y+z) constrained to the function g(x,y,z)=xyz36 .

Frame an auxiliary function F for the given function f(x,y,z) subject to g(x,y,z)=0 .

F(x,y,λ)=f(x,y,z)+g(x,y,z)=x+2(3y+z)+λ(xyz36)

The above expression is called the Lagrange function while λ be the Lagrange multipliers.

Differentiate F(x,y,λ)=x+2(3y+z)+λ(xyz36) with respect to x,

ddx(F(x,y,λ))=ddx(x+2(3y+z)+λ(xyz36))Fx=ddx(x+2(3y+z))+ddx(λ(xyz36))=1+yzλ

Differentiate F(x,y,λ)=x+2(3y+z)+λ(xyz36) with respect to y,

ddy(F(x,y,λ))=ddy(x+2(3y+z)+λ(xyz36))Fy=ddy(x+2(3y+z))+ddy(λ(xyz36))=6+λxz

Differentiate F(x,y,λ)=x+2(3y+z)+λ(xyz36) with respect to z,

ddz(F(x,y,

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach
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Ch. 8.1 - Let f(u, v) = (u2 + v2)euv2. Compute f(0, 1), f(1,...Ch. 8.1 - Let g(r, s, t) = res/t. 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How is...Ch. 8.6 - Suppose g1 and g2 are continuous and g1 (x) g2...Ch. 8.6 - PRINTED BY: f0988ba0c4e090b@ placeholder....Ch. 8.6 - What is the average value of f(x, y) over the...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 34-41, find the volume of the solid...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - Population Density of a Coastal Town The...Ch. 8.6 - Average Population Density of a Coastal Town Refer...Ch. 8.6 - Population Density Of A City The population...Ch. 8.6 - Average Profit The Country Workshops total weekly...Ch. 8.6 - Average Price of Land The rectangular region R...Ch. 8.6 - In Exercises 53-56, determine whether the...Ch. 8.6 - In Exercises 53-56, determine whether the...Ch. 8.6 - In Exercises 53-56, determine whether the...Ch. 8.6 - In Exercises 53-56, determine whether the...Ch. 8 - Fill in the blanks. 1. The domain of a function f...Ch. 8 - Fill in the blanks. 2. If the function f has rule...Ch. 8 - Fill in the blanks. 3. The graph of a function f...Ch. 8 - Fill in the blanks. 4. The trace of the graph...Ch. 8 - Fill in the blanks. 5. The partial derivative fx...Ch. 8 - Fill in the blanks. 6. The number fx(a, b)...Ch. 8 - Fill in the blanks. 7. A function f(x, y) has a...Ch. 8 - Fill in the blanks. 8. A critical point of f(x, y)...Ch. 8 - Fill the blanks. 9. By plotting the points...Ch. 8 - Fill in the blanks. 10. The method of Lagrange...Ch. 8 - Fill in the blanks. 11. If f(x, y) is continuous...Ch. 8 - Fill in the blanks. 12. The integral Rf(x,y)dA is...Ch. 8 - Let f(x,y)=xyx2+y2. Compute f(0, 1), f(1, 0), and...Ch. 8 - Let f(x, y) = f(x,y)=xey1+lnxy. Compute f(1, 1),...Ch. 8 - Leth (x, y, z) = xyez + xy. Compute h( 1, 1, 0),...Ch. 8 - Find the domain of the function f (u, v) = uuv.Ch. 8 - Find the domain of the function f (x, y) = xyx+y.Ch. 8 - Find the domain of the function f(x,y)=xy+y1xCh. 8 - Find The domain of the function...Ch. 8 - In Exercises 8-11, sketch the level curves of the...Ch. 8 - In Exercises 8-11, sketch the level curves of the...Ch. 8 - In Exercises 8-11, sketch the level curves of the...Ch. 8 - In Exercises 8-11, sketch the level curves of the...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 12-21, compute the first partial...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - In Exercises 22-27, compute the second-order...Ch. 8 - Let f(x, y, z) = x3 + y2z + xy2z + 3xy 4z....Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 29-34, find the critical point(s) of...Ch. 8 - In Exercises 35-38, use the method of Lagrange...Ch. 8 - In Exercises 35-38, use the method of Lagrange...Ch. 8 - Find the maximum and minimum values of the...Ch. 8 - Find the maximum and minimum values of the...Ch. 8 - In Exercises 39-42, evaluate the double integrals....Ch. 8 - In Exercises 39-42, evaluate the double integrals....Ch. 8 - In Exercises 39-42, evaluate the double integrals....Ch. 8 - In Exercises 39-42, evaluate the double integrals....Ch. 8 - In Exercises 43 and 44, find the volume of the...Ch. 8 - In Exercises 43 and 44, find the volume of the...Ch. 8 - Find the average value of The function f(x, y) =...Ch. 8 - IQs The IQ (intelligence quotient) of a person...Ch. 8 - Revenue Functions A division of Ditton Industries...Ch. 8 - Demand for CD Players In a survey concluded by...Ch. 8 - Average Daily TV-Viewing Time The following data...Ch. 8 - Female Life Expectancy at 65 The projections of...Ch. 8 - Mobile Phone Use in China The number of mobile...Ch. 8 - Maximizing Revenue Odyssey Travel Agencys monthly...Ch. 8 - Maximizing the Weigh is of Fish A pond is slacked...Ch. 8 - COBB-DOUGLAS PRODUCTION FUNCTIONS The production...Ch. 8 - COBB-DOUGLAS PRODUCTION FUNCTION Show that the...Ch. 8 - Find the domain of f(x,y)=x+y(1x)(2y)Ch. 8 - Find the first and second partial derivatives of...Ch. 8 - Find the relative extrema, if any, of f(x, y) =...Ch. 8 - Find the equation of the least-squares line for...Ch. 8 - Use the method of Lagrange multipliers to find the...Ch. 8 - Evaluate R(1xy)dA, where R is the region bounded...

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