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Question 1030027: Please HELP with homework problem!!
A dietitian designs a special dietary supplement using two different foods. Each ounce of food X contains 12 units of calcium, 10 units of iron, and 20 units of vitamin B. Each ounce of food Y contains 15 units of calcium, 20 units of iron and 12 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin B.
a) Write a system of inequalities describing the different amounts of food X and food Y that can be used.
b) Sketch a graph of the region in part (a)
c) Find two solutions to the system and interpret their meanings in the context of the problem.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = the number of ounces of food X.
let y = the number of ounces of food Y.
the total units of calcium have to be greater than 300.
the total units of iron have to be greater than 280.
the total units of vitamin B have to be greater than 300.
food X contains 12 units of calcium, 10 units of iron, and 20 units of vitamin B.
food Y contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B.
your constraint equations are:
12x + 15y >= 300
solve for y in this equation and it becomes y >= (300 - 12x) / 15
this means that 12 units of calcium * the number of ounces of food X plus 15 units of calcium * the number of ounces of food Y has to be greater than or equal to 300 units of calcium total.
10x + 20y >= 280
solve for y in this equation and it becomes y >= (280 - 10x) / 20
this means that 10 units of iron * the number of ounces of food X plus 20 units of iron * the number of ounces of food Y has to be greater than or equal to 280 units of iron total.
20x + 12y >= 300
solve for y in this equation and it becomes y >= (300 - 20x) / 12
this means that 20 units of vitamin B * the number of ounces of food X plus 12 units of vitamin B * the number of ounces of food Y has to be greater than or equal to 300 units of vitamin B total.
you also have some unmentioned constraints.
those are that x >= 0 and y >= 0.
this is because the number of ounces of food X and the number of ounces of food Y cannot each be negative.
you would need to graph the equality portion of these equations and then shade in the area of the inequality portion of these equations.
you would get a graph that looks like this, if you did it correctly.
the shaded region is the area on the graph that satisfies all of the inequalities.
it's the area that is above and on the graph of:
12x + 15y = 300
10x + 20y = 280
20x + 12y = 300
x = 0
y = 0
you only shade the area that meets all of the requirements.
that's called the feasible region.
the corner points of the feasible region give you the maximum or minimum value of the objective function.
since you don't have an objective function, these points tell you the minimum value that satisfies the constraints.
for example:
the point (0,25) is a corner point of the feasible region.
x = 0 and y = 25.
the equation of the line that it is on is the equation of 20x + 12y <= 300
how to i know?
solve for y in that equation and you get y <= (300 - 20x) / 12
when x = 0, that equation becomes y <= 25.
that's the point (0,25)
evaluate the equation of 20x + 12y >= 300 when x = 0 and y = 25 and you get:
20*0 + 12*25 <= 300 which results in 300 <= 300.
each of the points identified on the graph would satisfy the minimum value of the constraint equations.
those constraints are calcium >= 300, iron >= 280, vitamin B >= 300
best i can do for now.
if you have further questions, send me an email.
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