Inequalities question

Question

The Hits on a Shoestring music company is planning its next month’s work.
The company makes CDs of both rock and rap music.

It costs the company an average of $15,000 to produce a rock CD and an
average of $12,000 to produce a rap CD. Also, it takes about 18 hours to
produce a rock CD and about 25 hours to produce a rap CD.

The company can afford to spend up to $150,000 on product next month.
Also, the company agreed to spend at least 175 hours on production.

Hits on a Shoestring earns $20,000 in profit for each rock CD it produces
and $30,000 in profit for each rap CD it produces. But the company
recently promised its distributor that it would not release more rap music
than rock, based on the wishes of the company president.

The company needs to decide how many of each type of CD to make.
Note: It can make a fraction of a CD next month and finish it the month
after.

a. Express the constraints as inequalities, using x to represent the
number of rock CDs and y to represent the number of rap CDs.

b. Write an algebraic expression to present the profit in terms of x and y.

c. Graph the constraints to make the feasible region.

d. Determine how many rock and rap CDs Hit on Shoestring should
product to maximize its profits for next month.








Before submitting your work, have you...
___ answered all of the questions?
___ clearly showed your work?
___ explained why you took the mathem

Answer 1

 

Hi Sad,

 

a. Set of constraints (Eq. 1 through Eq. 5 shown below)

 

     x ≥ 0     (Eq. 1)

 

     y ≥ 0     (Eq. 2)

 

     15,000x + 12,000y ≤ 150,000     (Eq. 3)

 

     18x + 25y ≤ 175     (Eq. 4)

 

     x ≥ y     (Eq. 5)

 

b. Profit = 20,000x + 30,000y

 

c. Solving the set of systems, Eq. 1 though Eq. 5 shown above,

 

    we can identify the feasible region as a triangle

 

    of which three vertices are (x, y) = (0, 0), (9.92, 0), and (4.07, 4.07).

 

d. Out of those three points,

 

    (4.07, 4.07) maximizes the profit.

 

    As the company can make a fraction of a CD next month

 

    and finish it the month after,

    it should produce 5 rock CDs and 5 rap CDs.