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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.6, Problem 42E
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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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Biology Repeat Exercise 41 for a salt-content model given by

S =   0.01 x 2 y 2 z 2 .

Biology A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The proportion of salt in this medium is given by

S = 12xyz where x, y, and z are the amounts (in liters) of the three nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions x, y, and z cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

To determine

To calculate: The quantity of each nutrient that solution should be used to minimize the cost of medium solution s=0.01x2y2z2.

Explanation

Given information:

The proportion of salt in the medium is given by;

s=0.01x2y2z2

The medium must have 13% salt and the nutrient solution x, y and z cost is $1, $2, & $3 per litre respectively.

Formula used:

The standard equation is;

If f(x,y,z) be the function to minimize and g(x,y,z) be the constraint then

F(x,y,z)=f(x,y,z)−λg(x,y,z)

Where λ are Lagrange multiplier.

Calculation:

Consider the given equation,

f(x,y,z)=1×x+2xy+3λzf(x,y,z)=x+2y+3z

Now, consider the given function where s=0.01x2y2z2 must be 13% .

0.01x2y2z2=131000.01x2y2z2=0.13x2y2z2=0.130.01x2y2z2=13

Now, the constraint is g(x,y,z)=x2y2z2−13

Consider the standard equation;

F(x,y,z)=f(x,y,z)−λg(x,y,z)=x+2y+3z−λ(x2y2z2−13)

Now, differentiate the above equation with respect to x, y, z and λ

Fx(x,y,z,λ)=1−2λxy2z2Fy(x,y,z,λ)=2−2λx2yz2Fz(x,y,z,λ)=3−2λx2y2zFλ(x,y,z,λ)=(x2y2z2−13)

Equating the above each equation to zero to get the critical points as;

1−2λxy2z2=0

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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Evaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, fin...

Calculus (MindTap Course List)

Expand Your Knowledge: Geometric Mean When data consist of percentages, ratios, compounded growth rates, or oth...

Understanding Basic Statistics

Research results suggest a relationship between the TV viewing habits of 5-year-old children and their future p...

Statistics for The Behavioral Sciences (MindTap Course List)

Evaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 43. f(...

Single Variable Calculus

In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circu...

Probability and Statistics for Engineering and the Sciences

a. Use arrows to establish a one-to-one correspondence between V = {a, e, i} and M {3, 6, 9}. b. How many diffe...

Mathematical Excursions (MindTap Course List)

In Exercises 1728, use the logarithm identities to obtain the missing quantity.

Finite Mathematics

For Problems 1-50, solve each equation. Objective 1 3(2x1)=2(4x+7)

Intermediate Algebra

In Problem 8-10, solve the inequality and graph the solution

Mathematical Applications for the Management, Life, and Social Sciences

Find all values of x in the interval [0, 2] that satisfy the equation. 72. 2 + cos 2x = 3 cos x

Single Variable Calculus: Early Transcendentals

Find the remaining trigonometric functions of , if the terminal side of lies along the line y=2x in QIII.

Trigonometry (MindTap Course List)

In O not shown, OA is a radius, AB is a diameter, and AC is a chord. a How does OA compare to AB? b How does AC...

Elementary Geometry For College Students, 7e

Exercises Find an isomorphism from the octic group D4 in Example 12 of this section to the group G=I2,R,R2,R3,H...

Elements Of Modern Algebra

Determine the range of the following temperature readings from the oven at Bon Appétit Bakery.

Contemporary Mathematics for Business & Consumers

Fill in each blank: 275in2=cm2

Elementary Technical Mathematics

Finding a Derivative In Exercises 17-42, find the derivative of the function. g(x)=arcsin3xx

Calculus: Early Transcendental Functions (MindTap Course List)

Using the Trapezoidal Rule and Simpsonss Rule In Exercises 15-24, approximate the definite integral using the T...

Calculus of a Single Variable

Let A be a matrix of size mn, and let B be a matrix of size st. Find conditions on m, n, s and t such that both...

Finite Mathematics for the Managerial, Life, and Social Sciences

Draw the graph of 4x-3y=9 by first changing the equation to Slope-Intercept Form.

Elementary Geometry for College Students

is the binomial series for:

Study Guide for Stewart's Multivariable Calculus, 8th

Finding a Derivative In Exercises 47-52, find the derivative of the function. g(t)=t2et

Calculus

Maximum RevenueA company manufactures running shoes and basketball shoes. The total revenue (in thousands of do...

Calculus (MindTap Course List)

Finding Surface AreaIn Exercises 316, find the area of the surface given by z=f(x,y) that lies above the region...

Multivariable Calculus

Sometimes, Always, or Never: If 0 ≤ an ≤ bn for all n and diverges, then converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Divide: 97.126 + 0.53. Round the answer to 4 decimal places.

Mathematics For Machine Technology

Practice Solve each problem. Setting fares A bus company has 3000 passengers daily, paying a 25fare. For each n...

College Algebra (MindTap Course List)

Identity and describe the component-analysis design and describe the circumstances in which it is used.

Research Methods for the Behavioral Sciences (MindTap Course List)

Explain the difference between a linear relationship and a monotonic relationship, and identify which correlati...

Research Methods for the Behavioral Sciences (MindTap Course List)

In exercise 5, the owner of Showtime Movie Theaters, Inc., used multiple regression analysis to predict gross r...

Statistics for Business & Economics, Revised (MindTap Course List)

Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the range and interquartile range.

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Suppose that the researchers who carried out the experiment described in the previous exercise thought that sex...

Introduction To Statistics And Data Analysis

Use Hamiltons Method to apportion the legislative seats in Exercise 2. 2. A small country consists of three sta...

Mathematics: A Practical Odyssey

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