Linear Programming word problem
Full Question:
Madison is preparing a wilderness backpacking trip on which she plans to eat a combination of cereals and nuts. The recommended daily requirement is at least 300 calories and 2.4 ounces of protein. Each ounce of cereal supplies 120 calories and 0.04 ounces of protein. Each ounce of nuts provides 60 calories and 0.12 ounces of protein. If the cereal costs 20 cents per ounce and each ounce of nuts costs 30 cents, what combination should she plan to eat in order to keep the cost to a minimum?
There are so many variables that are confusing me. Here is the list I have so far:
Cereal
nuts
ounces
calories
protein
cost': 30 cents, 20 cents
I need to make two inequalities and then isolate the variables. From there I know how to graph. Its setting up the inequalities that is giving me trouble. Any help is very much appreciated.
To solve this linear programming word problem, we can start by defining our variables and formulating the objective function and the constraints.
Let's define our variables:
Let x be the number of ounces of cereal.
Let y be the number of ounces of nuts.
Objective function: We want to minimize the total cost of the combination of cereals and nuts.
The cost of cereal per ounce is given as 20 cents, and the cost of nuts per ounce is given as 30 cents. Hence, the objective function can be written as:
Cost = 0.20x + 0.30y
Constraints:
1) Calorie Requirement: The recommended daily requirement is at least 300 calories.
Each ounce of cereal supplies 120 calories, and each ounce of nuts provides 60 calories. Therefore, the calorie constraint can be written as:
120x + 60y ≥ 300
2) Protein Requirement: The recommended daily requirement is at least 2.4 ounces of protein.
Each ounce of cereal supplies 0.04 ounces of protein, and each ounce of nuts provides 0.12 ounces of protein. Therefore, the protein constraint can be written as:
0.04x + 0.12y ≥ 2.4
Non-Negative Constraints:
x ≥ 0 (The number of ounces of cereal should be non-negative)
y ≥ 0 (The number of ounces of nuts should be non-negative)
Now, you have your objective function and your constraints in terms of the variables x and y. You can graph these constraints on a two-dimensional coordinate plane and find the feasible region. The point within the feasible region that minimizes the objective function will give you the optimal combination of cereals and nuts that Madison should plan to eat in order to keep the cost to a minimum.
Note: To graph the inequalities, you can plot the lines for each constraint and shade the region of the graph that satisfies all the constraints. The feasible region will be the shaded area. The optimal solution will then be the point within this region that minimizes the objective function.