1. Distribution. An oil refinery in Tulsa sells 50% of its production to a Chicago distributor, 20% to a Dallas distributor, and 30% to an Atlanta distributor. Another refinery in New Orleans sells 40% of its production to the Chicago distributor, 40% to the Dallas distributor, and 20% to the Atlanta distributor. A third refinery in Ardmore sells the same distributors 30%, 40%, and 30% of its production. The three distributors received 219,000, 192,000, and 144,000 gal of oil, respectively. How many gallons of oil were produced at each of the three plants?
2. Chemistry. When carbon monoxide (CO) reacts with oxygen (O2), carbon dioxide
(CO2) is formed. This can be written as CO + (1/2) O2 = CO2 and as a matrix equation. If we form a 2 x 1 column matrix by letting the first element be the number of carbon atoms and the second element be the number of oxygen atoms, then CO would have the column matrix
Similarly, O2 and CO2 would have the column matrices
and , respectively.
a. Use the Gauss-Jordan method to find numbers x and y (known as stoichiometric numbers) that solve the system of equations
Compare your answers to the equation written above.
b. Repeat the process for xCO2 = yH2 = zCO = H2O, where H2 is hydrogen, and H2O is water. In words, what does this mean?
3. In Chapter 2 we wrote a system of linear equations using matrix notation. We can
do the same thing for the system of linear inequalities in this chapter.
a. Find matrices A, B, C, and X such that the maximization problem in Example 1 of Section 4.1 can be written as
Maximize CX
subject to: AX ≤ B
with X ≥ 0
(Hint: Let B and X be column matrices, and C a row matrix).
b. Show that the dual of the problem in part a can be written as
Minimize YB
subject to: YA ≥ C
with Y ≥ 0
where Y is a row matrix.
c. Show that for any feasible solutions X and Y to the original and dual problems, respectively, CX ≤ YB. (Hint: Multiply both sides of AX ≤ B by Y on the left. Then substitute for YA.)
d. For the solution X to the maximization problem and Y the dual, it can be shown that
CX = YB
is always true. Verify this for Example 1 of Section 4.1. What is the
significance of the value in CX (or YB)?
4. Suppose a family plans 6 children, and the probability that a particular child is a girl is ½. Find the probability that the 6-child family has the following children.
At least 4 girls
5. Baseball. The number of runs scored in 16,456 half-innings of the 1986 National League Baseball season was analyzed by Hal Stern. Use the following table to answer the following questions.
a. What is the probability that a given team scored 5 or more runs in any given half-inning during the 1986 season?
b. What is the probability that a given team scored fewer than 2 runs in any given half-inning of the 1986 season?
c. What is the expected number of runs that a team scored during any given half-inning of the 1986 season? Interpret this number.
Chemical Effectiveness. White flies are devastating California crops. An area infested with white flies is to be sprayed with a chemical which is known to be 98% effective for each application. Assume a sample of 1000 flies is checked.
6. Find the approximate probability that at least 975 of the flies are killed in one application.