You know that 
, and 
, Find the
maximum value of 
and the coordinates of the point where it
occurs. Is this the same as the corner where y-3x has its maximum
value? (Justify your answer by drawing the graph and showing the
values of these functions at each corner point.)
Let 
and
Compute 
.
Compute the inverse of 
or show that it has no inverse. Show and indicate all your row
operations. No credit for the correct answer without work shown.
(This problem is a modification of
one from the text.)
Ted's Toys makes toy airplanes, boats, and cars. The net profit on each airplane is three dollars, the net profit on each car is four dollars, and the net profit on each boat is five dollars. The materials used are plastic, wood strips, and steel. Each airplane uses 100 grams of plastic, 10 inches of wood strips, and 200 grams of steel. Each boat uses 50 grams of plastic, 100 inches of wood strips, and 50 grams of steel; and each car uses 50 grams of plastic and 150 grams of steel. If Ted's has on hand 10,500 grams of plastic, 1500 inches of wood strips, and 25,500 grams of steel, how many planes, boats, and cars should be made to maximize the total profits from this production run.
Define variables and set up this problem as a linear programming problem. Write down each inequality with a sentence of explanation as to what fact makes it part of the problem. (There are six inequalities in all, although three of the six can be justified by the same sentence.) Also, write down the expression for the profit to be maximized in terms of your variables.
Do not attempt to solve this problem, since it involves three
variables and is difficult to graph.
A certain virus has the following epidemological characteristic: A
person can be immune to the virus, healthy but susceptible to the
virus, or infected with the virus. Of those healthy but susceptible,
one percent will become infected in any given year. Of those infected,
60 % will recover within a year, and be immune to further
infection. Additionally, another 10% of the infected population
will recover but will still be susceptible to reinfection. A person
who is immune remains immune. No one becomes immune without being
infected first.
- Using the three states:
State 1: Healthy but susceptible.
State 2: Infected.
State 3: Healthy and immune.Set up a Markov chain describing this situation. Draw a transition diagram and write down a transition matrix.
What is the probability that a person infected at present will be in
the infected population three years from now? (You must compute the
cube of the transition matrix to answer this, or else use a tree
diagram.)
