   Chapter 7.6, Problem 42E ### 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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# 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 byS = 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=0.01x2y2z2mustbe13%.

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

Now, the constraint is g(x,y,z)=x2y2z213

Consider the standard equation;

F(x,y,z)=f(x,y,z)λg(x,y,z)=x+2y+3zλ(x2y2z213)

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

Fx(x,y,z,λ)=12λxy2z2Fy(x,y,z,λ)=22λx2yz2Fz(x,y,z,λ)=32λx2y2zFλ(x,y,z,λ)=(x2y2z213)

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

12λxy2z2=0

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