151
Linear Mathematics
Test 3
Name (please print)_____________________________________________
Show your work!
Solve problems 1 and 2 using the geometric approach.
1. An oil company requires 9000, 12,000, and 26,000 barrels of high-, medium-,
and low-grade oil, respectively. It owns two oil refineries, A and B. Refinery A
produces 100, 300, and 400 barrels of high-, medium-, and low-grade oil,
respectively, per day; refinery B produces 200, 100, and 300 barrels, respectively.
How many days should each refinery to be run to meet the requirements and
minimize costs in each of the following cases?
(a) Each refinery costs $20,000 per day to operate.
(b) Refinery A costs $30,000 per day to operate, and refinery B, $20,000.
2. A baker has 150, 90, and 150 units of ingredients A, B, and C, respectively. A
loaf of bread requires 1, 1, and 2 units of A, B, and C, respectively; a cake requires
5, 2, and 1 units of A, B, and C, respectively. How many of each should be baked
in order to maximize the gross income in each of the following cases?
(a) A loaf of bread sells for $1.40 and a cake for $3.20.
(b) A loaf of bread sells for $1.80 and a cake for $3.20.
Use simplex method to solve problem 3.
3. A company uses machines L, M, and N to produce products P, Q, and R. Each
item of product P requires 5 hours on machine L, and 4 hours each on machines M
and N. Each item of product Q requires 2 hours on L, 5 hours on M, and 3 hours
on Q, while each item of R requires 2 hours on L and 6 hours on each M and N.
All three machines operate a total of 40 hours per week. If the company makes a
profit of $4 on each item of P, and $3 and $4.50 on each item of Q and R,
respectively, how many items of each product should be produced each week in
order to maximize the total profit?
Use duality to solve problem 4.
4. Minimize z = 8x1 + 4x2 + 16x3 subject to
2x1 + 2x2 + 3x3  16
3x1 + x2 + 4x3  14
3x1 + x2 + 5x3  12
x1 , x2 , x3  0