Question

The staff dietician at the California Institute of Trigonometry has to make up a meal with 600 calories, 20 grams of protein, and 200 milligrams of vitamin C. There are three food types to choose from: rubbery jello, dried fish sticks, and mystery meat. They have the following nutritional content per ounce. Jello Fish Sticks Mystery Meat Calories 10 50 200 Protein 1 3 .2 Vitamin C 30 10 o (a) Make a mathematical model of the dietician\'s problem with a system of three linear equations. (b) Find an approximate solution (accurate to within 10%)

Answer 1

Solutions/Explanation: (a) Let the amount of rubbery jello=x ounce, amount of dried fish sticks=y ounce and amount of mystery meat=z ounce.

The given ingredients have following specifications:

(1) rubbery jello has: Calories=10 Cal/ounce, Protien=1 gram/ounce, Vitamin C= 30 mg/ounce

(2) dried fish sticks: Calories=50 Cal/ounce, Protien=3 grams/ounce, Vitamin C=10 mg/ounce

(3) Mystery meat: Calories=200 Cal/ounce, Protien=0.2 gram/ounce, Vitamin C=0 mg/ounce,

Since, total calories=600. So,

10x+50y+200z=600 --------------------------------------------------- (1)

Since, total Protien=20 g. So,

x+3y+0.2z=20 ------------------------------------------------- (2)

Since, total Vitamin C=200 mg. So,

30x+10y+0z=200 ------------------------------------------------ (3)

So, Equations 1, 2 and 3 are the mathematical models of the dietician's problem with a system of three linear equations.

(b) Since, we have three equatiosn:

10x+50y+200z=600

OR x+5y+20=60 ---------------------------------------- (4)

From eqn 2, x+3y+0.2z=20 ---------------------------------------- (5)

From eqn 3. 30x+10y=200

OR 3x+y=20 -------------------------------------- (6)

For solving the eqn 4, 5 and 6 for x, y and z,

Multiplying eqn 5 by 100,

100x+300y+20z=2000 ------------------------------------ (7)

Now, subtract eqn 4 from eqn 7

99x+295y=1940 --------------------------------------- (8)

Now, multiplying eqn 6 by 295. we have,

885x+295y=5900 ---------------------------------------- (9)

Now, subtract, eqn 8 from eqn 9, we get,

786x=3960

x=3960/786

  x=5.038 --------------------------------------- (10)

Now, substituting x=5.038 into eqn 6, we get

y=4.886 -------------------------------------------(11)

Now, substitute eqn 10 and 11 into eqn 4, we have

z=1.526

Hence, the amount of rubbery jello=5.038 ounce, dried fish stick=4.886 ounce and mystery meat=1.526 ounce.