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.
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 is:
R = 4x + 3y
b) To find the vertices for the feasible region, we need to consider the constraints given:
Constraint 1: Time constraint
It takes Melanie 30 minutes to make a bracelet and 15 minutes to make a keychain. She has 3 hours (180 minutes) per week to spend making the products.
So, the time constraint can be written as:
30x + 15y ≤ 180
Constraint 2: Materials constraint
Melanie only has enough materials to make 10 items per week.
So, the materials constraint can be written as:
x + y ≤ 10
To find the vertices, we graph the feasible region formed by these two constraints:
The feasible region is bounded by the lines:
30x + 15y = 180 (time constraint)
x + y = 10 (materials constraint)
By graphing the lines and finding the points where they intersect, we can determine the vertices of the feasible region.
c) To determine the number of each item Melanie should make every week in order to maximize the money raised, we need to find the vertex that maximizes the objective function R = 4x + 3y within the feasible region.
d) The maximum amount of money Melanie could raise each week can be found by substituting the values of the vertex that maximizes the objective function R = 4x + 3y into the objective function equation.