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.
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 that would allow Melanie to maximize her money raised can be represented as:
R = 4x + 3y
b) To find the vertices for the feasible region, we need to consider the constraints:
1) Time constraint: It takes Melanie 30 minutes to make a bracelet and 15 minutes to make a keychain, and she has a total of 3 hours (180 minutes) per week. Therefore, the time constraint can be represented as:
30x + 15y ≤ 180
2) Material constraint: Melanie only has enough materials to make 10 items each week. Therefore, the material constraint can be represented as:
x + y ≤ 10
In order to find the vertices, we can solve the system of inequalities formed by the constraints:
30x + 15y ≤ 180
x + y ≤ 10
x ≥ 0, y ≥ 0
By graphing the inequalities, we find the vertices of the feasible region are:
(0, 0), (0, 10), (6, 0), (4, 4)
c) To find the number of each item Melanie should make every week in order to maximize the amount of money she can raise, we can substitute the vertices into the objective function and find the values that result in the maximum value for R = 4x + 3y.
For the vertices:
(0, 0): R = 4(0) + 3(0) = 0
(0, 10): R = 4(0) + 3(10) = 30
(6, 0): R = 4(6) + 3(0) = 24
(4, 4): R = 4(4) + 3(4) = 28
Comparing the values, we see that the maximum value for money raised is R = 30, which occurs when Melanie makes 0 bracelets and 10 keychains.
Therefore, Melanie should make 0 bracelets and 10 keychains every week in order to maximize the amount of money she can raise.
d) The maximum amount of money Melanie could raise each week is $30.