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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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Nutrition The number of grams of your favorite ice cream can be modeled by

G   = ( x , y , z )   =   0.05. x 2 +   0.16 x y   +   0.25 z 2 where x is the number of fat grams, y is the number of carbohydrate grams, and z is the number of protein grams. Find the maximum number of grams of ice cream you can eat without consuming more than 400 calories. Assume that there are 9 calories per fat gram. 4 calories per carbohydrate gram, and 4 calories per protein gram.

To determine

To calculate: The maximum number of grams of ice cream, that can be eaten without consuming more than 400 calories if the provided function is G(x,y,z)=0.05x2+0.16xy+0.25z2.

Explanation

Given Information:

Given the function that models the number of grams of the favourite ice cream is as,

G(x,y,z)=0.05x2+0.16xy+0.25z2

Where x is number of fat grams

y is carbohydrate grams

z is number of protein grams

Also, there are 9 calories per fat gram, 4 calories per carbohydrate gram, 4 calories per

protein gram

Formula used:

Step 1: If x is the number of grams of the substance consumed and a is the number of calories

per gram in the substance x, the calories consumed is ax.

Step 2: If f(x,y) has a maximum or a minimum subjected to constraint g(x,y)=0, then it

Will occur at one of the critical points defined by function F as

F(x,y,λ)=f(x,y)λg(x,y)

This method is the of Lagrange multipliers where λ is a Lagrange multiplier value of f(x,y),

The following equations are to be solved,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fz(x,y,λ)=0

f(x,y) is evaluated at every point as obtained above:

Calculation:

Consider the function,

G(x,y,z)=0.05x2+0.16xy+0.25z2

According to primary concept and given information,

C(x,y,z)=9x+4y+4z=400

Where C(x,y,z) is a calorie modelling function.

According to constraint of not having more than 400 calories,

C(x,y,z)=9x+4y+4z=400

According to secondary formula,

H(x,y,z,λ)=G(x,y,z)λ(F(x,y,z))H(x,y,z)=0.05x2+0.16xy+0.25z2λ(9x+4y+4z400)

As the constraint function.

Where λ is Lagrange multiplier.

To find maximum,

Hx(x,y,z,λ)=0Hy(x,y,z,λ)=0Hz(x,y,z,λ)=0Hλ(x,y,z,λ)=0

Here, H(x,y,z)=0

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