corn and squash dinner to provide at least 3 grams of protein and cost no more than 36 cents per serving.
1 ounce of creamed corn provgides 1/2 gram of protein and costs 4 cents.
1 ounce of squash supplies 1/4 gram of protein and costs 3 cents.
you need at least 2 ounces of corn.
you also need at least 2 ounces of squash as well.
you need to keep the total amound of ounces of corn and squash to a minimum.
the total cost must be less than 36 cents per serving.
let x equal the number of ounces of corn.
let y equal the number of ounces of squash.
your objective function would be x + y.
that's what you want to minimize.
your cost will be 4x + 3y and it has to be less than or equal to 36 cents.
your constraint function for cost will be 4x + 3y <= 36
you will need at least 2 ounces of corn and 2 ounces of squash.
your constraint functions for weight will be:
x >= 2
y >= 2
the total amount of protein must be at least 3 grams.
your constraint function for protein will be 1/2 * x + 1/4 * y >= 3
summarizing.
objective function is to minimize x + y
constraint functions are:
x >= 2
y >= 2
4x + 3y <= 36
1/2 * x + 1/4 * y >= 3
you would graph these functions and find the area of the curve that satisfies all the constraints.
my graph would look like this:
i would graph the equalities and then look for the region of feasibility that satisfies the inequalities.
the equations i graphed are:
x = 2
y = 2
4x + 3y =36
1/2 * x + 1/4 * y = 36
the region of feasibility is the region on the graph that satisfies the inequalities.
that area would be where:
x >= 2
y >= 2
4x + 3y <= 36
1/2 * x + 1/4 * y >= 3
the arrows from the lines indicate where the region of feasibility lies.
anywhere in that region between those lines is the feasible region.
all the constraints are satisfied within that region.
i would then look for the corners of the region of feasibility and evaluate the objective function at those corners.
the maximum or minimum value for the feasible region would be found at those corners.
the corners of the region of feasibility are:
(2,9+1/3)
(2,8)
(5,2)
(7.5,2)]
the objective function is x + y which leads to:
11+1/3
10
10
9.5
at each of those respective corner points.
the minimum number of ounces is at (7.5,2) where the total number of ounces are 7.5 + 2 = 9.5.
the constraints are satisfied at that point because:
x = 7.5 which is >= 2
y = 2 which is >= 2
4x + 3y = (4 * 7.5) + (3 * 2) = 30 + 6 = 36 which is <= 36
1/2 * x + 1/4 * y = (1/2 * 7.5) + (1/4 * 2) = 3.25 + .5 = 3.75 which is >= 3
you were able to minimize the number of ounces while still satisfying the constraints.
your solution should be 7.5 ounces of corn and 2 ounces of squash will be the minimum amount of total ingredients that you need to satisfy the requirements.