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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 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 =   12 x y z 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 amount of nutrient solution should be used to minimize the cost of the culture medium if the proportion of salt in the medium is given by s=12xyz.

Explanation

Given Information:

The proportion of salt in the medium is given by,

s=12xyz

Where x, y, z is the amount (in liters) of three nutrient so:

The medium must be 13% and the cost of x,yandzare$1,$2&$3 respectively.

Formula used:

Step 1: First find the function and constant which have to minimize.

Step 2: If f(x,y,z) be the function and g(x,y,z)be the constant then find new function

f(x,y,z)=f(x,y,z)λg(x,y,z).

Where λ the Lagrange multiplier.

Step 3: Equate 1st partial derivatives of f(x,y,z) with respect to x,y,z,λ with zero to find the minimum value and find the points x, y and z.

Calculation:

Consider the provided information:

Here the function s=12xyz must be 13%.

g(x,y,z)=12xyz0.13

Now, the function is which have to minimize.

f(x,y,z)=x+2y+3z=x+2y+3z

Consider the standard equation,

f(x,y,z)=x+2y+3zλ(12xyz0.13)

Now, the partial derivative of f(x,y,z) with respect to x, y, z and λ,

fx(x,y,z,λ)=112yzxfy(x,y,z,λ)=212xzλfz(x,y,z,λ)=312xyλfλ(x,y,z,λ)=(12xyz0

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