Question

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?

Answer 1

Melanie can maximize her money raised by making 6 bracelets and 12 keychains every week. The maximum amount of money she could raise each week is $60.

Step-by-step explanation:

a) The objective function that would allow Melanie to maximize her money raised is R = 4x + 3y, where x is the number of bracelets made and y is the number of keychains.

b) To find the vertices for the feasible region, we need to consider the constraints. Melanie can only spend 3 hours making the products, and it takes 30 minutes to make a bracelet and 15 minutes to make a keychain. This means she can make a maximum of 6 bracelets (3 hours = 180 minutes, 180 minutes / 30 minutes per bracelet = 6 bracelets) and 12 keychains (3 hours = 180 minutes, 180 minutes / 15 minutes per keychain = 12 keychains). Thus, the vertices of the feasible region are (0,0), (6,0), and (0,12).

c) To maximize the amount of money she can raise, Melanie should make 6 bracelets and 12 keychains every week.

d) The maximum amount of money Melanie could raise each week would be $4 * 6 (bracelets) + $3 * 12 (keychains) = $24 + $36 = $60.

Answer 2

a) The objective function is R = 4x + 3y. b) The vertices of the feasible region are determined by solving the time and materials constraints. c) To maximize money raised, Melanie should make as many bracelets as possible. d) The maximum amount of money Melanie can raise can be found by substituting the optimal values into the objective function.

Step-by-step explanation:

a) The objective function that would allow Melanie to maximize her money raised can be written as follows:

R = 4x + 3y

where R is the total money raised, x is the number of bracelets made, and y is the number of keychains made.

b) To find the vertices of the feasible region, we need to consider the constraints. The time constraint can be written as:

30x + 15y ≤ 180

where 180 represents the total number of minutes Melanie has to spend making the products per week. The materials constraint can be written as:

x + y ≤ 10

where 10 represents the total number of items Melanie can make each week. Solving these two constraints together will give us the vertices of the feasible region.

c) To maximize the amount of money Melanie can raise, she should make as many bracelets as possible and fill the remaining spots with keychains. Let's solve the constraints to find the optimal values:

30x + 15y ≤ 180

x + y ≤ 10

Solving these two equations will give us the values of x and y.

d) To find the maximum amount of money Melanie can raise, we can substitute the optimal values of x and y into the objective function:

R = 4x + 3y

Calculating this will give us the maximum amount of money Melanie can raise each week.