alright so one more example here of a
word problem using linear programming to
find a in this case of minimum suppose
you've got a rancher he's got two types
of food that he's going to feed his
cattle brand x brand Y suppose you know
for each serving of food has to have 60
grams of protein and 30 grams of fat
brand a has 15 grams of protein and 10
grams of fat
it costs eighty cents per unit Brandi
has 20 grams of protein and five grams
of fat and costs fifty cents per unit
we want to know how much of each type of
food this rancher would have to use in
order to minimize these costs
ok so let's see a couple things here
we're trying to minimize his costs
and let's again let's define our
variables your first so Brand X how about
we let X represent the we'll say the
number of units
of Brand X and the same thing will
simply just let little y represent the
number of units of brand Y so here we
are trying to minimize his costs and
again we said where to go
let's see so brand a we said cost eighty
cents per unit so its cost would be 0.8
for each unit of Brand X plus we said
brand Y cost fifty cents per unit so
plus 0.5
y this is going to be we're going to
try to minimize our constraints in this
case
well X would have to be greater than or
equal to 0 and y would have to be
greater than or equal to 0
well let's see we've got some
restrictions
ok we know there has to be so many grams
of protein and there has to be so many
grams of fat
let's see so if we look at the stuff
that relates protein so you need 60
grams of protein total brand a has 15
grams of protein
it says brained he has 20 grams of
protein so let's see here so
to me it says we're going to get 15
grams of protein per each unit of Brand
X and we're going to get an additional
20 grams of protein per each for each
unit from brand Y and again we want that
to be at least 60
ok so this is kind of the inequality
that relates the the demands from
getting enough protein
ok now we also have to have 30 grams of
fat and we said you get let's see brand
A has 10 grams of fat and it looks like
Brand B has five grams of fat
so we want to get at least 30 grams of
fat in our diet
it says again Brand A has let's see 10
grams per 10 grams of fat per each unit
Brand B has five grams of fat
per each unit and I think I think that's
everything we need now I think we've got
all of our all of our constraints and
now we can go about graphing the region
and figure out our corner points and
then use that to define our minimum
value
ok so I'm going to graph 15x plus 20y
equals 60 and again to do this i think
i'm just going to find x and
y-intercepts to be I think that's the
easiest way in this case
so if we substitute in x equals 0
we'll just get 20y equals 60 if we
divide both sides by 20 we'll get a
y-value of three
likewise if we plug in y equals 0 we'll
have 15x equals 60 if we divide both
sides by 15 will get x equals 4
alright so we connect those two points
there
looks like we would get that line again
we're trying to satisfy this original
inequality 15x plus 20y greater than or
equal to 60 i would take the test .
00 and if we plug that in we'll get
15 times 0 plus 20 times 0 is that
greater than or equal to 60
well we're going to get 0 on the left 0
is definitely not greater than or equal
to 60
so that tells me we'd have to shade
above that line to get the region that
satisfies that inequality again since X
is greater than or equal to 0 and also y
is greater than or equal to 0
I know that we're going to be trapped up
here in the you know the top right
quadrant
let's graph are other line as well
so 10 x + 5 y equals 30 so 10 x + 5 y
equals 30
again I think I would do the same thing
just find x and y-intercepts if we plug
in x equals 0 we'll get 5y equals 30 or y
equals 6
if we plug in y equals 0 will get 10 x
equals 30 if we divide both sides by 10
will get x equals 3
alright so let's see
3 zero and then we have four five six
so if we connect those two
those two points
we're going to get our other line and
again the same thing we can take our
test point we're trying to satisfy 10 x + 5
y greater than or equal to 30 same thing
we could take maybe this test point of the
origin if we plug in 0 and 0. Zero for X and
0 for y
well then I'm thinking is 0 greater than
or equal to 30 again definitely not so
that means we would have to shade above
this above and to the right of this line
to get the region that satisfies that
inequality
so it looks like in total to me the the
overlap of all of our region's we would
get actually a region that extends off
forever and ever and ever
it's infinite in extent
and that makes sense you know because
you could always use you know a million
units of Brand X and a million units of
brand why that would certainly give you
the amount of protein in the matter of
fact that you need in your diet
probably a little more than what you
need but now all i'm going to need are
these corner points so we know that
whatever corner points is 4 comma 0
another one is zero comma six the only one I
missing here is this point of
intersection between these two lines so
we're gonna have to figure out the point
of intersection
so let's see we've got 15 x plus 20 y
equals 60 and i'm going to use 10 x + 5
y equals 30
I think what i'm going to do is well
let's see what would be the easiest
thing to do here
i think i'm just going to use
elimination by addition and i'm going to
cancel out the Y's
so what I'm going to do is I'm going to
multiply both sides of my second
equation by negative 4 i'm gonna leave
the first equation alone
15x plus 20 y equals 60
we'll get negative 40 x minus 20 y
equals negative 120 if we do our
elimination by addition
let's see it looks like we get so if we
add 15 x and negative 40 x that's going
to give us negative 25 x
that's going to be equal to negative 60
if we divide both sides by negative 25 by
negative 25
let's see that's going to give us
positive let's see 12
let's see so we can divide the top by
five we can divide the bottom by five so
that's going to give us
what is that - two and two-fifths or 2.4 and
in this case you know i'm going to
assume he can use you know he doesn't
have to use a whole number of units of
food you know maybe he can you know
chop it up or grind it up or whatever so
you know we're not going to make the
restriction that he has to use a whole
number of units of food so 2.4 to me
that's going to be certainly something
reasonable
let's see and we now have to figure out
so we know the x coordinate here is 2.4
now we just need to go back and
figure out the y value that goes with it
again we can use any of our lines
i'm going to use this original 10x plus
5y equals 30 so if we use that to solve
for y will get 10 x 2.4 plus 5y equals
30
well 10 x 2.4 is going to be 24
and if we subtract 24 from both sides
will get 5y equals 6 and then we can
divide will get 6 over 5.        6 over 5 would
be 1 and 1/5th or 1.2
so it looks like this point of
intersection is 2.4 , 1.2
so now we're in business we've got all
our corner points here
now i just need to go back to my cost
function so our cost function was point 8x
plus point 5y again we're trying to minimize
this and the points are going to use our
06
we've got this point to point 4 comma
1.2 and then we've got this point 4
comma 0
alright so that's just some plugging and
chugging so if we use 06 our cost is
going to be well point eight times 0 which
is 0 plus point 5 x y and why has the value
of six so that's going to give us the
value 3 i'm going to plug in 4 comma
zero next because well that's easier to
do if we plug in 4 we'll get point 8 times
4
Plus point 5 x 0     point 8 times 4 is going to give
us a cost of 3.2 and last but not least
if we use 2.4 comma 1.2        We'll have C equals point eight
times the x value of 2.4 plus point five
times the Y value of 1.2
let's see point eight times 2.4 that's
1.92 one half of 1.2
would be .6 if we add those together
we're going to get just that
let me make sure one more time so 1.92
plus point six
so let's see that's going to give us 2.52
let's just use my brain here
and I did something wrong and it didn't
look right
so that looks like we're going to get
the value 2.52 and in this case again
we're trying to find a minimum cost i
think we've got it now we've got the
cost of three cost of 3.2 the cost of
2.52 well certainly this is going to be
our minimum cost so it looks like to me
he should use two point four units of
Brand X and 1.2 units of brand Y
to get our a minimum cost but at the
same time it would meet all the
nutritional requirements for his cattle
under using these inequalities
