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 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 material 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 to maximize the money raised, R, would be:
R = 4x + 3y
b) To find the vertices for the feasible region, we need to consider the constraints given:
- Melanie can only spend a maximum of 3 hours or 180 minutes making the products per week.
- It takes her 30 minutes to make a bracelet and 15 minutes to make a keychain.
- She only has enough material to make 10 items each week.
Let's solve these constraints graphically:
- Let x represent the number of bracelets and y represent the number of keychains.
- Melanie can spend a maximum of 180 minutes making the products, so we have the constraint: 30x + 15y ≤ 180.
- Melanie can make a maximum of 10 items, so we have the constraint: x + y ≤ 10.
Graphing these constraints, we get a feasible region defined by the shaded area below:
10 | ┌──────┐
| │ │
| │ ▓ │
| │ ▓▓ │
| │ ▓▓▓ │
y | │▓▓▓▓ │
| │▓▓▓▓▓ │
| │▓▓▓▓▓▓│
| │▓▓▓▓▓▓│
─────┼───────────
| x
The vertices of the feasible region are:
(0, 0)
(0, 10)
(6, 4)
(10, 0)
c) To maximize the amount of money Melanie can raise, we need to find the vertex that yields the highest R value (4x + 3y). From the feasible region, the vertex (6, 4) would maximize the money raised.
Therefore, Melanie should make 6 bracelets and 4 keychains every week to maximize the amount of money she can raise.
d) To find the maximum amount of money Melanie could raise each week, substitute the values into the objective function:
R = 4(6) + 3(4) = 24 + 12 = $36.
Thus, the maximum amount of money Melanie could raise each week is $36.