Melanie is trying to raise money for her local animal shelter and decides to start selling friendship bracelets and keychains She raises $4 with the sale of each bracelet and $3 for each keychainIt takes her about 30 minutes to make every bracelet , and about 15 minutes for each keychain Because she is still in school full time, Melanie only has 3 hours to spend making the products per weekAlso, she only has enough materials to make 10 items each to necessary work in order to receive credit R represent the total money raisedWrite the objective function that would allow maximize her money raised, where x is the number of bracelets made and y is the number of keychains vertices for the feasible region Make sure to include either a graph or a description of how you found your answers How many of each should Melanie make every week in order to maximize the amount of money she can raise What would be the maximum amount of money she could raise each week ?

Question

Melanie is trying to raise money for her local animal shelter and decides to start selling friendship bracelets and keychains She raises $4 with the sale of each bracelet and $3 for each keychainIt takes her about 30 minutes to make every bracelet , and about 15 minutes for each keychain Because she is still in school full time, Melanie only has 3 hours to spend making the products per weekAlso, she only has enough materials to make 10 items each to necessary work in order to receive credit R represent the total money raisedWrite the objective function that would allow maximize her money raised, where x is the number of bracelets made and y is the number of keychains vertices for the feasible region Make sure to include either a graph or a description of how you found your answers How many of each should Melanie make every week in order to maximize the amount of money she can raise What would be the maximum amount of money she could raise each week ?

Answer 1

To write the objective function, we need to maximize the total money raised (R). The total money raised can be calculated by multiplying the number of bracelets made (x) by the amount raised per bracelet ($4) and adding it to the product of the number of keychains made (y) and the amount raised per keychain ($3).

Therefore, the objective function is:
R = 4x + 3y

Now, let's consider the constraints:

1) Time constraint:
Melanie only has 3 hours to spend making the products per week, which is equivalent to 180 minutes. Since making a bracelet takes 30 minutes and making a keychain takes 15 minutes, the time constraint can be represented as:
30x + 15y ≤ 180

2) Materials constraint:
Melanie only has enough materials to make 10 items in total. Therefore, the materials constraint can be represented as:
x + y ≤ 10

3) Non-negativity constraint:
Since we cannot make a negative number of bracelets or keychains, the non-negativity constraint can be represented as:
x ≥ 0
y ≥ 0

Next, we need to graph these constraints to find the feasible region.

Graphing the time constraint:
Let's solve the equation 30x + 15y ≤ 180 for y and graph it.
30x + 15y ≤ 180
15y ≤ -30x + 180
y ≤ -2x + 12

Graphing the materials constraint:
Let's solve the equation x + y ≤ 10 for y and graph it.
y ≤ -x + 10

Now, we can find the feasible region by graphing the two inequalities together.

The shaded area represents the feasible region. Let's find the vertices of the feasible region:

- Vertex 1: (0, 0)
- Vertex 2: (0, 10)
- Vertex 3: (4, 6)
- Vertex 4: (10, 0)

To determine the number of bracelets and keychains Melanie should make in order to maximize the amount of money raised, we need to evaluate the objective function at each vertex and find the highest value.

- Vertex 1: R = 4(0) + 3(0) = 0
- Vertex 2: R = 4(0) + 3(10) = 30
- Vertex 3: R = 4(4) + 3(6) = 34 + 18 = 52
- Vertex 4: R = 4(10) + 3(0) = 40

The maximum amount of money Melanie could raise each week is $52. To achieve this, she should make 4 bracelets and 6 keychains.