Use the information below to answer each part.
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 keychain. It 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 week. Also, she only has enough materials to make 10 items each week.
Make sure to include all necessary work in order to receive full credit.
a) Let R represent the total money raised. Write the objective function that would allow Melanie to maximize her money raised, where x is the number of bracelets made and y is the number of keychains.
b) List all vertices for the feasible region. Make sure to include either a graph or a description of how you found your answers.
c) How many of each item should Melanie make every week in order to maximize the amount of money she can raise?
d) What would be the maximum amount of money she could raise each week?
a) The objective function would be: R = 4x + 3y
b) To find the vertices of the feasible region, we need to consider the constraints mentioned in the problem:
- Time constraint: 30 minutes per bracelet and 15 minutes per keychain, with a total of 3 hours (180 minutes) available. So the time constraint can be written as: 30x + 15y ≤ 180
- Material constraint: Melanie only has enough materials to make 10 items per week. So the material constraint can be written as: x + y ≤ 10
To find the vertices, we can solve these two inequalities as a system of linear equations:
30x + 15y ≤ 180
x + y ≤ 10
First, let's solve the equation x + y ≤ 10 for y:
y ≤ 10 - x
Now substitute this expression for y in the first inequality:
30x + 15(10 - x) ≤ 180
30x + 150 - 15x ≤ 180
15x ≤ 30
x ≤ 2
So the possible values for x are 0, 1, or 2.
For x = 0:
10 - x = 10 - 0 = 10
The coordinates (0, 10) satisfy both inequalities.
For x = 1:
10 - x = 10 - 1 = 9
The coordinates (1, 9) also satisfy both inequalities.
For x = 2:
10 - x = 10 - 2 = 8
The coordinates (2, 8) satisfy both inequalities as well.
Therefore, the vertices of the feasible region are (0, 10), (1, 9), and (2, 8).
c) To maximize the amount of money Melanie can raise, we need to evaluate the objective function R = 4x + 3y at each vertex.
For (0, 10):
R = 4(0) + 3(10) = 0 + 30 = $30
For (1, 9):
R = 4(1) + 3(9) = 4 + 27 = $31
For (2, 8):
R = 4(2) + 3(8) = 8 + 24 = $32
So Melanie should make 2 bracelets and 8 keychains every week to maximize the amount of money she can raise.
d) The maximum amount of money Melanie could raise each week is $32.