WEBVTT
1
00:00:02.540 --> 00:00:06.809 A:middle L:90%
all right, so we have an optimization problem and
2
00:00:06.809 --> 00:00:11.099 A:middle L:90%
which we have a production formula in. Our goal
3
00:00:11.109 --> 00:00:16.730 A:middle L:90%
is to maximize the production when the cost of the
4
00:00:16.739 --> 00:00:21.539 A:middle L:90%
production can be no more than$40,000. So when
5
00:00:21.539 --> 00:00:23.890 A:middle L:90%
we want to do is talk about our variables and
6
00:00:23.890 --> 00:00:26.379 A:middle L:90%
talk about what we need to maximize staff, our
7
00:00:26.440 --> 00:00:30.570 A:middle L:90%
cost function, so are variables. Are Axum y
8
00:00:30.780 --> 00:00:34.770 A:middle L:90%
where X represents units of labor and why represents units
9
00:00:34.770 --> 00:00:40.700 A:middle L:90%
of capital we need to create for this a cost
10
00:00:40.700 --> 00:00:45.850 A:middle L:90%
function. So we're told that the, um,
11
00:00:46.640 --> 00:00:52.689 A:middle L:90%
the problem is$80 a unit of labor. So
12
00:00:52.689 --> 00:00:57.240 A:middle L:90%
if it is$80 per unit of flavor, then
13
00:00:57.240 --> 00:01:00.170 A:middle L:90%
it is 80 x plus. We're told that is
14
00:01:00.170 --> 00:01:04.760 A:middle L:90%
$150 per unit of capital. So that's 150.
15
00:01:04.760 --> 00:01:07.930 A:middle L:90%
Why? And then again, we're told that we
16
00:01:07.930 --> 00:01:14.700 A:middle L:90%
have$40,000. So our firm Ken's been no more
17
00:01:14.700 --> 00:01:19.409 A:middle L:90%
than$40,000 at 80 x plus 150. Why?
18
00:01:22.540 --> 00:01:25.950 A:middle L:90%
What we're going to do then, is take that
19
00:01:25.950 --> 00:01:30.040 A:middle L:90%
cost function and we are going to get the problems
20
00:01:30.049 --> 00:01:36.379 A:middle L:90%
of one variable using are constructed. So I'm gonna
21
00:01:36.379 --> 00:01:45.069 A:middle L:90%
take the 40,000 equal to 80 x plus 150.
22
00:01:45.069 --> 00:01:48.560 A:middle L:90%
Why? And we're going to solve that problem for
23
00:01:48.560 --> 00:01:52.930 A:middle L:90%
why or for X Now, in this case,
24
00:01:52.939 --> 00:01:55.359 A:middle L:90%
I chose to solve for X, but you could
25
00:01:55.359 --> 00:01:57.939 A:middle L:90%
also solve her way. So I am going to
26
00:01:57.939 --> 00:02:04.129 A:middle L:90%
subtract the 150 wife from both sides and divide by
27
00:02:04.640 --> 00:02:15.849 A:middle L:90%
80 and just a quick, simple simplifying of that
28
00:02:16.340 --> 00:02:23.960 A:middle L:90%
. We get 500 minus 15 8th Why and then
29
00:02:23.960 --> 00:02:27.550 A:middle L:90%
what I'm going to do is I'm going to substitute
30
00:02:28.139 --> 00:02:32.400 A:middle L:90%
this piece of information in here in do my f
31
00:02:32.409 --> 00:02:37.250 A:middle L:90%
of X by that I was given to start.
32
00:02:42.740 --> 00:02:45.370 A:middle L:90%
So we're going to replace the X with that red
33
00:02:45.370 --> 00:02:50.490 A:middle L:90%
function, so that now creates three and in terms
34
00:02:50.490 --> 00:02:54.490 A:middle L:90%
off, why 500 minus 15? It's why,
35
00:02:54.939 --> 00:03:00.770 A:middle L:90%
to the 1/3 power times wide of the 2/3.
36
00:03:01.939 --> 00:03:05.669 A:middle L:90%
Why did I put this in terms of one variable
37
00:03:05.669 --> 00:03:06.969 A:middle L:90%
? Well, being able to put it in terms
38
00:03:06.969 --> 00:03:09.909 A:middle L:90%
of one variable allows me to find my derivative without
39
00:03:09.909 --> 00:03:15.039 A:middle L:90%
needing to use implicit differentiation. It also allows me
40
00:03:15.039 --> 00:03:19.840 A:middle L:90%
to find it derivative and find the zeros of that
41
00:03:19.840 --> 00:03:23.039 A:middle L:90%
derivative by being able to find the zeros. I'll
42
00:03:23.039 --> 00:03:24.870 A:middle L:90%
be able to find the why values at which the
43
00:03:24.870 --> 00:03:29.759 A:middle L:90%
point could potentially be maximized. So what we're gonna
44
00:03:29.759 --> 00:03:31.569 A:middle L:90%
do is we're gonna take our new function. We're
45
00:03:31.569 --> 00:03:37.340 A:middle L:90%
going to find its derivative. So after prime,
46
00:03:38.860 --> 00:03:40.659 A:middle L:90%
and in order to do that, we need to
47
00:03:40.659 --> 00:03:44.750 A:middle L:90%
use the product room because we have two things raised
48
00:03:44.750 --> 00:03:46.509 A:middle L:90%
to a power and we're actually going to use the
49
00:03:46.509 --> 00:03:50.689 A:middle L:90%
chain role within the product roll. So the product
50
00:03:50.689 --> 00:04:00.419 A:middle L:90%
rule says you have the first. Now we're gonna
51
00:04:00.419 --> 00:04:02.069 A:middle L:90%
do is take it and multiply it by the derivative
52
00:04:02.069 --> 00:04:04.990 A:middle L:90%
of the second. So the derivative of wider the
53
00:04:04.990 --> 00:04:12.169 A:middle L:90%
2/3 is 2/3. Why to the negative 1/3 and
54
00:04:12.169 --> 00:04:15.449 A:middle L:90%
I were going toe. Add that to the second
55
00:04:15.439 --> 00:04:18.470 A:middle L:90%
, which was, why did the 2/3 and multiply
56
00:04:18.470 --> 00:04:24.420 A:middle L:90%
it by the derivative of the first? So that's
57
00:04:24.430 --> 00:04:34.930 A:middle L:90%
three times 1/3. Keep the inside. Now we
58
00:04:34.930 --> 00:04:39.259 A:middle L:90%
have to subtract one from the exponents on the outside
59
00:04:39.259 --> 00:04:44.170 A:middle L:90%
, so you get negative 2/3 and then we're going
60
00:04:44.170 --> 00:04:46.129 A:middle L:90%
to multiply that by the derivative of the insides,
61
00:04:46.129 --> 00:04:50.980 A:middle L:90%
a derivative of 500 cancels and then derivative of negative
62
00:04:50.980 --> 00:05:00.139 A:middle L:90%
15 8th Why is negative? 15 8th now?
63
00:05:00.139 --> 00:05:02.800 A:middle L:90%
We're just gonna do some simple cleaning up three times
64
00:05:02.800 --> 00:05:09.100 A:middle L:90%
, 1/3 cancel three and 2/3. The threes are
65
00:05:09.100 --> 00:05:15.129 A:middle L:90%
going to cancel, and then we are going to
66
00:05:15.129 --> 00:05:26.110 A:middle L:90%
simplify this. So that gives me to times the
67
00:05:26.110 --> 00:05:32.649 A:middle L:90%
quantity 500 minus 15 8th supply to the 1/3 all
68
00:05:32.649 --> 00:05:39.209 A:middle L:90%
ever. Why? To the 1/3 minus, We're
69
00:05:39.209 --> 00:05:45.459 A:middle L:90%
gonna pull the fraction out and front 15 8th and
70
00:05:45.459 --> 00:05:49.199 A:middle L:90%
then we have why to 2/3. And that is
71
00:05:49.199 --> 00:05:56.660 A:middle L:90%
going to be all over 500 minus 15 pates.
72
00:05:56.660 --> 00:06:01.240 A:middle L:90%
Why? To the 2/3. In order to make
73
00:06:01.240 --> 00:06:05.459 A:middle L:90%
this problem easier, we're gonna get a common denominator
74
00:06:05.639 --> 00:06:10.470 A:middle L:90%
between the two. So that means that we need
75
00:06:10.470 --> 00:06:14.459 A:middle L:90%
to multiply both fractions by the other denominators. So
76
00:06:14.459 --> 00:06:18.439 A:middle L:90%
over here, two times eight is 16. 1/3
77
00:06:18.439 --> 00:06:21.569 A:middle L:90%
time's 2/3 will give us one. So we get
78
00:06:21.569 --> 00:06:29.610 A:middle L:90%
500 minus 15 eights. Why minus? And then
79
00:06:29.610 --> 00:06:30.800 A:middle L:90%
if you multiply over here, why did the 2/3
80
00:06:30.800 --> 00:06:34.759 A:middle L:90%
times why do the 1/3 gives you why CF minus
81
00:06:34.759 --> 00:06:40.199 A:middle L:90%
15? Why all over this new denominator of eight
82
00:06:40.199 --> 00:06:46.199 A:middle L:90%
to the 1/3 500 minus 15 8th Why? To
83
00:06:46.199 --> 00:06:55.430 A:middle L:90%
the 2/3. And there you have our derivatives of
84
00:06:55.439 --> 00:07:00.750 A:middle L:90%
our far are crossed production function sort of combined.
85
00:07:00.139 --> 00:07:03.180 A:middle L:90%
So what we can do is use this to now
86
00:07:03.180 --> 00:07:06.949 A:middle L:90%
find the points of which production could be maximized.
87
00:07:08.439 --> 00:07:12.649 A:middle L:90%
So maximization can occur at the zeros of the derivative
88
00:07:13.639 --> 00:07:15.500 A:middle L:90%
. So are gonna dio is set that derivative equal
89
00:07:15.500 --> 00:07:24.560 A:middle L:90%
to zero. So in doing so, we get
90
00:07:24.569 --> 00:07:33.879 A:middle L:90%
zero equals 16 times 500 minus 15 8th Why bias
91
00:07:33.889 --> 00:07:38.060 A:middle L:90%
15? Why now? We don't even worry about
92
00:07:38.060 --> 00:07:41.129 A:middle L:90%
setting the denominator equal to zero because that will give
93
00:07:41.129 --> 00:07:43.949 A:middle L:90%
me the points at which the derivative does not exist
94
00:07:44.439 --> 00:07:46.480 A:middle L:90%
, which are vertical Assam tubes which would not be
95
00:07:46.480 --> 00:07:50.000 A:middle L:90%
points of maximums or minimums. They wouldn't be an
96
00:07:50.000 --> 00:07:55.430 A:middle L:90%
extra Emma. So what I did at this point
97
00:07:55.430 --> 00:08:01.459 A:middle L:90%
was distributed thesixties een in. So I multiplied 16
98
00:08:01.459 --> 00:08:09.889 A:middle L:90%
by 500 and that gave me 8000. And then
99
00:08:09.889 --> 00:08:16.079 A:middle L:90%
I multiplied 16 by 15/8 and what was nice about
100
00:08:16.079 --> 00:08:18.970 A:middle L:90%
16 times eight. Was it divided evenly and give
101
00:08:18.970 --> 00:08:22.930 A:middle L:90%
you two. So you got negative 30 y minus
102
00:08:22.930 --> 00:08:31.860 A:middle L:90%
15. Why? And then at this point,
103
00:08:31.860 --> 00:08:39.169 A:middle L:90%
we were able to get that are wise combined,
104
00:08:39.169 --> 00:08:46.149 A:middle L:90%
say about 8000 minus 45. Why? And then
105
00:08:46.149 --> 00:08:50.620 A:middle L:90%
I'm able to solve for why and I get the
106
00:08:52.039 --> 00:09:16.139 A:middle L:90%
why is equal to 177.7 repeating. Okay, So
107
00:09:16.139 --> 00:09:18.610 A:middle L:90%
if we want to use this line short to determine
108
00:09:18.610 --> 00:09:22.039 A:middle L:90%
that this is a maximum, what will we have
109
00:09:22.049 --> 00:09:26.850 A:middle L:90%
to do is put the 1177.7 on or sign shirt
110
00:09:28.240 --> 00:09:31.600 A:middle L:90%
. We're using our F prime equation to determine this
111
00:09:31.820 --> 00:09:33.230 A:middle L:90%
, and what you would do is just kind of
112
00:09:33.230 --> 00:09:39.399 A:middle L:90%
plug in numbers to have prime to determine that it
113
00:09:39.399 --> 00:09:41.980 A:middle L:90%
is positive to the left and negative to the right
114
00:09:43.480 --> 00:09:48.309 A:middle L:90%
, meaning that the original would be increasing to decreasing
115
00:09:52.440 --> 00:09:54.019 A:middle L:90%
. So again, if you just kind of quickly
116
00:09:54.019 --> 00:09:58.340 A:middle L:90%
plugged in numbers to your original equation of the derivative
117
00:09:58.440 --> 00:10:03.059 A:middle L:90%
and I actually plugged my numbers in to this form
118
00:10:03.159 --> 00:10:07.360 A:middle L:90%
right here, so I'd say 8000 minus. And
119
00:10:07.360 --> 00:10:11.440 A:middle L:90%
then again, you can pick kinda any number less
120
00:10:11.440 --> 00:10:16.120 A:middle L:90%
than that so you could do 8000 minus 45 times
121
00:10:16.129 --> 00:10:22.250 A:middle L:90%
100 and we know that 8000 minus 45 times 100
122
00:10:24.240 --> 00:10:28.250 A:middle L:90%
would be 3500 which would give us a positive number
123
00:10:30.639 --> 00:10:31.250 A:middle L:90%
. And then if you did this same and you
124
00:10:31.250 --> 00:10:35.379 A:middle L:90%
pick a number larger than 1 77 So I picked
125
00:10:35.389 --> 00:10:41.570 A:middle L:90%
200. We can see easily that 8000 and 45
126
00:10:41.570 --> 00:10:46.129 A:middle L:90%
times 200 is going to give me in negative 1000
127
00:10:46.450 --> 00:10:50.269 A:middle L:90%
. So there's the negative. And that means that
128
00:10:50.269 --> 00:10:54.549 A:middle L:90%
the original is increasing, then decreasing. Which proves
129
00:10:54.549 --> 00:10:58.659 A:middle L:90%
that this is the max value for why so then
130
00:10:58.659 --> 00:11:01.340 A:middle L:90%
what we're gonna do to find X is plugged that
131
00:11:01.340 --> 00:11:05.149 A:middle L:90%
value back in. So we have that X is
132
00:11:05.149 --> 00:11:11.259 A:middle L:90%
equal to 500 minus 15 AIDS. Why? And
133
00:11:11.259 --> 00:11:15.850 A:middle L:90%
that was from our simplified cost function and that we're
134
00:11:15.850 --> 00:11:18.750 A:middle L:90%
just going to say OK, 500 minus 15 8
135
00:11:20.190 --> 00:11:30.159 A:middle L:90%
times 1 77.7 and that gives me 166.7. So
136
00:11:30.159 --> 00:11:35.340 A:middle L:90%
we obviously cannot have 0.7 of a product. So
137
00:11:35.350 --> 00:11:46.879 A:middle L:90%
what that means is that 177 units of capital and
138
00:11:46.889 --> 00:12:05.549 A:middle L:90%
166 units of labor well a maximize the company's production
139
00:12:13.340 --> 00:12:26.600 A:middle L:90%
at a cost less than or equal to 40,000.
140
00:12:28.320 --> 00:12:30.649 A:middle L:90%
And again, if you wanted to check your work
141
00:12:30.649 --> 00:12:33.059 A:middle L:90%
and make sure your numbers were accurate, you could
142
00:12:33.059 --> 00:12:39.490 A:middle L:90%
find the cost. At 1 66 1 77 by
143
00:12:39.490 --> 00:12:41.820 A:middle L:90%
just plugging it in. So it would be 80
144
00:12:41.820 --> 00:12:48.350 A:middle L:90%
times when 66 plus 1 50 times 1 77 And
145
00:12:48.350 --> 00:12:52.779 A:middle L:90%
you get that your costs is 39,000, 830.
146
00:12:52.710 --> 00:12:54.330 A:middle L:90%
So we didn't.