-your school club decides to hold a fundraiser by selling trail mix, and you are in chrage of making the mix. You plan to offer two mixes, Country Blend and Premium Mix, each sold in one pound bags. Each pound of Country Blend consists of 1/2 pound of toasted oats, 1/4 pound of peanuts, and 1/4 pound of raisins. Each pound of Premium Mix consists of 1/4 pound of toasted oats, 1/4 pound of peanuts, and 1/2 pound of raisins. You have available to use at most 40 pounds of oats, 22 pounds of peanuts, and 35 pounds of raisins.
1. Model the situation above by letting X represent the # of pounds of Country Blend and Y represent the # of pounds of Premium Mix. Your algebraic model should be a system of five inequalities.
2. Graph it.
3. You sell the trail mix for $5 per pound for Country Blend and $7 per pound for Premium Mix. How many bags of each type of mix should you make it order to maximize your income? (The maximum income must occur at one of the vertices of the graph.)
4. Using the answer from #3, what will be your club's income if all the bags of mix are sold?
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1. To model the situation, we need to represent the constraints for the availability of oats, peanuts, and raisins as inequalities. Let X represent the number of pounds of Country Blend, and Y represent the number of pounds of Premium Mix.
a) Oats constraint: The total amount of oats used in both mixes should be at most 40 pounds.
This can be represented as: 1/2X + 1/4Y ≤ 40
b) Peanuts constraint: The total amount of peanuts used in both mixes should be at most 22 pounds.
This can be represented as: 1/4X + 1/4Y ≤ 22
c) Raisins constraint: The total amount of raisins used in both mixes should be at most 35 pounds.
This can be represented as: 1/4X + 1/2Y ≤ 35
d) Non-negativity constraint: The number of pounds of each mix cannot be negative.
This can be represented as: X ≥ 0 and Y ≥ 0
e) Integer constraint: The number of pounds of each mix must be whole numbers since we can't sell fractions of a pound.
This can be represented as: X is an integer and Y is an integer
2. To graph it, plot the inequalities on a coordinate plane.
a) Plot the line 1/2X + 1/4Y = 40 (the equality part of the oats constraint) by finding two points that satisfy this equation and drawing a line through them.
b) Shade the area below this line because we want the total amount of oats to be at most 40 pounds.
c) Repeat the process for the remaining inequalities.
3. To find the maximum income, we need to analyze the vertices of the feasible region formed by the intersection of the shaded areas. Calculate the income at each vertex by multiplying the number of pounds of Country Blend (X) and Premium Mix (Y) by their respective prices. Compare the income at each vertex and identify the maximum.
4. Using the answer from step 3, once you know how many bags of each type of mix to make, multiply the number of bags by the price per pound and add them to find the total income.