NAME:

Do the following problems in the text or as listed below. Be sure to show your work. Follow the directions listed with each problem.

  1. Set up the matrix equation to solve this problem. (You do not have to solve the problem, just set up the matrix equation.)

Transportation Ace Trucking Company has an order for three products, A, B, and C, for delivery. The table below gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of the products. If the carrier can carry 8000 cu ft and 12,400 lb and is insured for $52,600, how many units of each product can be carried?

Product A / Product B / Product C
Unit Volume (cu ft) / 24 / 20 / 40
Weight (lb) / 40 / 30 / 60
Value ($) / 150 / 180 / 200

Solution:

Let a be the units of product A, b be the units of product B and c be the units of product C.

Matrix equation will be

Use the simplex method to solve the linear program below. Solve using the simplex tool on the Hofstra web site. Copy/paste the linear program you used and the solution matrix from the Hofstra site into your test.

Advertising

The Laposta pasta Company has $12,000 available for advertising. The following Table gives the cost per ad and the numbers of people exposed to its ads in three different media (with numbers in thousands).

Ad Packages / Newspaper / Radio / TV
Cost / 2 / 2 / 4
Total Audience / 30 / 21 / 54
Working Mothers / 6 / 12 / 8

If the total available audience is 420,000, and if the company wishes to maximize the number of exposures to working mothers, how many ads of each type should it purchase?

Solution:

Let x be the Newspaper ad, y be the number of Radio ad and z be the number of TV ad.

Linear program will be

Maximize P = 6000x + 12000y + 8000z

Subject to

2000x + 2000y + 4000z ≤ 12000

30,000x + 21000y + 54000z ≤420,000

x,y,z ≥ 0

Tableau #1

x y z s1 s2 p

2000 2000 4000 1 0 0 12000

30000 21000 54000 0 1 0 420000

-6000 -12000 -8000 0 0 1 0

Tableau #2

x y z s1 s2 p

1 1 2 0 0 0 6

9000 0 12000 -21/2 1 0 294000

6000 0 16000 6 0 1 72000

Optimal solution is P = 72000; x = 0, y = 6, z = 0

To maximize the number of exposures to working mothers,

Number of Newspaper ad = 0

Number of Radio ad = 6

Number of TV ad = 0

  1. 625/20Find C /(3) and be sure to answer the question.

20.If the function for a commodity is

dollars

Solution:

Put x = 3

find the marginal cost units and tell what this predicts about the cost of producing 1 additional unit and 2 additional units.

Solution:

Marginal cost = C’(3) = $28.30

It means that cost would increase by approximately $28.30 on the production of one additional unit and increase by $56.60 for 2 additional units.

  1. 780/4Integrate the marginal cost function to get the cost function in terms of the constant K. Then use x = 20 when C(20) = 2000 to evaluate K. Then write the cost function using this value for K.

If the marginal cost for a product is , and the total cost of producing 20 units is $2,000, find the total cost function.

Solution:

C =

+ 50x +K

Given C(20) = 2000

Put x = 20

Thus,

  1. 836/26Change C to C = 250 + x2 + 5x. Use p as it is stated in the problem.

In problem 26, and are in dollars and is the number of units.

A monopoly has a total cost function for its product, which has demand function Find the consumer’s surplus at the point where the monopoly has maximum profit.

Solution:

Revenue R = xp =

Profit P = R – C

Put P’(x) = 0

= 0

P”(x) = -2x-6

P”(4.35) <0 so profit is maximum at x = 4.35

Consumer surplus = 34.99

= - 152.2065

= - 152.2065

= 37.23

Answer: 37.23

Bonus. (10 points) see below

Solve the matrix equation in Problem 1 above. Be sure to show the inverse matrix you used.

Solution:

Matrix equation will be

Here A =

Answer:

a = 100, b = 120, c = 80