Question 1155593: 9) A diet specialist tells patient to add oranges and apples to her diet for Vitamin A, Vitamin B, and Vitamin C. A kg (kilogram) of oranges contributes 8 grams of Vitamin C, 6 grams of Vitamin B, and 2 grams of Vitamin A, while a kg of apples contributes 3 grams of Vitamin A and 6 grams of Vitamin B. A kg of oranges costs $1.95 and a kg of apples costs $2.99. The patient has to meet a minimum requirement of 22 grams of Vitamin A, 33 grams of Vitamin B, and 16 grams of Vitamin C. She should avoid taking more than 73 grams of Vitamin A and 64 grams of Vitamin B to avoid side effects such as nausea. Formulate a linear programming problem to help the patient choose the ideal fruit mix for their diet.
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website! .
A diet specialist tells patient to add oranges and apples to her diet for Vitamin A, Vitamin B, and Vitamin C.
A kg (kilogram) of oranges contributes 8 grams of Vitamin C, 6 grams of Vitamin B, and 2 grams of Vitamin A, while
a kg of apples contributes 3 grams of Vitamin A and 6 grams of Vitamin B.
A kg of oranges costs $1.95 and a kg of apples costs $2.99.
The patient has to meet a minimum requirement of 22 grams of Vitamin A, 33 grams of Vitamin B, and 16 grams of Vitamin C.
She should avoid taking more than 73 grams of Vitamin A and 64 grams of Vitamin B to avoid side effects such as nausea.
Formulate a linear programming problem to help the patient choose the ideal fruit mix for their diet.
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Let x be the amount of oranges, and let y be the amount of apples for optimal diet (in kilograms).
Then the objective function to minimize is the price P = 1.95x + 2.99y dollars.
The constraints are
22 <= 2x + 3y <= 73 (the amount of vitamin A, in grams)
33 <= 6x + 6y <= 64 (the amount of vitamin B, in grams)
16 <= 8x (the amount of vitamin C, in grams)
x >= 0, y >= 0.
The setup is just completed.
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To see many problems solved by the Linear Programming method, look into the lesson
- Solving minimax problems by the Linear Programming method
in this site.
Also, look into the solutions of other similar problems in the archive to this forum under the links
https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1148776.html
https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1142000.html
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1137172.html
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1136382.html
https://www.algebra.com/algebra/homework/Coordinate-system/Coordinate-system.faq.question.1134444.html
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html
https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html
https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html
https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1123217.html (!)
https://www.algebra.com/algebra/homework/playground/test.faq.question.1112482.html
https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html
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In order for the formulation be precisely correct, from the common sense point of view,
it must say for what time period this diet is designed (a day, a week, and so on . . . )
It does not play any role in the setup and/or in the solution,
but is NECESSARY PART from the common sense point of view.
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