A store sells two types of nut mixtures. The standard mix contains 100 grams of cashews and 200 grams of peanuts and sells for $2. The deluxe mix contains 120 grams of cashews and 40 grams of peanuts and sells for $2.50. The store has 15kilograms of cashews and 20 kilograms of peanuts available. Based on past sales it needs to have at least as many standards as deluxe packages available. Represent algebraically (ie with equations and/or inequalities using defined variables) and geometrically (ie with a graph) the set of possible combinations of standard and deluxe packages the store could make.

Can someone help me with this question? I'm honestly confused because my teacher showed it to my class but I didn't understand.

Let s and d be how many standard and deluxe packages

Let c and d be the amounts of cashews and peanuts

Now figure how many grams of each nut will be needed for the packages
120s + 120d <= 15000
200s + 40d <= 20000
s >= d

You did not ask, but I assume you want to maximize revenue.
Try using this in your linear algebra methods.

Let's assume that the store sells x standard nut mixtures and y deluxe nut mixtures.

To represent the algebraic equations, we can consider the quantity constraints for cashews and peanuts.

For cashews:
The standard mix contains 100 grams of cashews, and the deluxe mix contains 120 grams of cashews. So, the cashew quantity constraint can be written as:
100x + 120y <= 15,000 (since there are 15 kilograms of cashews available, and 1 kilogram = 1000 grams)

For peanuts:
The standard mix contains 200 grams of peanuts, and the deluxe mix contains 40 grams of peanuts. So, the peanut quantity constraint can be written as:
200x + 40y <= 20,000 (since there are 20 kilograms of peanuts available, and 1 kilogram = 1000 grams)

To represent the inequality based on past sales, we need to have at least as many standards as deluxe packages available:
x >= y

To represent this geometrically, you can plot these inequalities on a graph. The x-axis represents the number of standard packages (x), and the y-axis represents the number of deluxe packages (y).

Plotting the cashew constraint (100x + 120y <= 15,000) on the graph, you can draw a line passing through points (0,125) and (150,0).

Plotting the peanut constraint (200x + 40y <= 20,000) on the graph, you can draw a line passing through points (0,500) and (100,0).

Finally, on the same graph, shade the region where x >= y because the store needs to have at least as many standards as deluxe packages.

The shaded region, which is the solution to the system of inequalities and the inequality, represents the set of possible combinations of standard and deluxe packages the store could make.

I hope this explanation helps! Let me know if you have any further questions.

Sure, I can help explain how to approach this question.

To represent the set of possible combinations of standard and deluxe packages algebraically, let's use variables to represent the number of standard and deluxe packages the store could make.

Let's call the number of standard packages "S" and the number of deluxe packages "D".

First, we know that the total grams of cashews used in the standard mix is 100 grams per standard package and 120 grams per deluxe package. So, the total grams of cashews used in the standard and deluxe packages can be represented as:

Total grams of cashews used = 100S + 120D

Similarly, the total grams of peanuts used in the standard and deluxe packages can be represented as:

Total grams of peanuts used = 200S + 40D

Next, we need to consider the available quantities of cashews and peanuts. The store has 15 kilograms of cashews, which is equal to 15,000 grams, and 20 kilograms of peanuts, which is equal to 20,000 grams. We can represent the available quantities as:

Available cashews = 15,000 grams
Available peanuts = 20,000 grams

We can now set up inequalities to represent the available quantities of cashews and peanuts. Since the store cannot use more than the available quantities, the inequalities are:

100S + 120D ≤ 15,000 (for cashews)
200S + 40D ≤ 20,000 (for peanuts)

Lastly, we need to consider the requirement that the store needs to have at least as many standard packages as deluxe packages available. This can be represented as:

S ≥ D

To represent the set of possible combinations graphically, each possible combination of standard and deluxe packages can be plotted on a graph with S on the x-axis and D on the y-axis. The inequalities mentioned above can be represented as shaded regions on the graph.

I hope this explanation helps you better understand the problem. Let me know if you have any further questions!