1
00:00:00,000 --> 00:00:00,000

2
00:00:00,000 --> 00:00:08,630
PROFESSOR: Welcome back
to recitation.

3
00:00:08,630 --> 00:00:11,510
In this video we'd like to do
another optimization problem.

4
00:00:11,510 --> 00:00:14,610
This one's a little bit harder
than the distance problem.

5
00:00:14,610 --> 00:00:16,750
So the question is
the following--

6
00:00:16,750 --> 00:00:19,650
consider triangles formed by
lines passing through the

7
00:00:19,650 --> 00:00:21,950
point x, h, 4-- sorry--

8
00:00:21,950 --> 00:00:24,390
the x-axis and the y-axis.

9
00:00:24,390 --> 00:00:26,680
Find the dimensions that
minimize area.

10
00:00:26,680 --> 00:00:28,010
So what does this fist
sentence mean?

11
00:00:28,010 --> 00:00:32,140
It really means use this
point to draw a line

12
00:00:32,140 --> 00:00:32,800
through this point--

13
00:00:32,800 --> 00:00:36,040
I'll give you an example, it's
kind of a wiggly line, but

14
00:00:36,040 --> 00:00:38,730
hopefully it looks like
a line to you--

15
00:00:38,730 --> 00:00:41,400
and it makes a triangle
with this line, the

16
00:00:41,400 --> 00:00:42,980
x-axis, and the y-axis.

17
00:00:42,980 --> 00:00:46,130
We can certainly calculate the
area of that triangle.

18
00:00:46,130 --> 00:00:48,470
So the problem is asking you to
find the dimensions of the

19
00:00:48,470 --> 00:00:53,197
triangle that minimize the area
with the constraint that

20
00:00:53,197 --> 00:00:56,330
the line, the hypotenuse goes
through the point 8, 4.

21
00:00:56,330 --> 00:00:58,010
I'm going to give you a couple
minutes to work on it.

22
00:00:58,010 --> 00:01:01,480
Why don't you pause video here
and then when you're ready,

23
00:01:01,480 --> 00:01:03,810
restart the video, I'll come
back, and I'll help you solve

24
00:01:03,810 --> 00:01:05,060
the problem.

25
00:01:05,060 --> 00:01:13,150

26
00:01:13,150 --> 00:01:14,460
Welcome back.

27
00:01:14,460 --> 00:01:16,860
So again, we're doing an
optimization problem.

28
00:01:16,860 --> 00:01:18,470
And we want to optimize--

29
00:01:18,470 --> 00:01:20,800
because it says minimize area,
we know the optimizing

30
00:01:20,800 --> 00:01:22,270
equation is area.

31
00:01:22,270 --> 00:01:24,580
So let's be very clear.

32
00:01:24,580 --> 00:01:26,810
Always, when you're doing these
problems, you have,

33
00:01:26,810 --> 00:01:28,860
again, as we've said previously,
you have a

34
00:01:28,860 --> 00:01:32,140
constraint equation and you have
an optimizing equation.

35
00:01:32,140 --> 00:01:35,860
The optimizing equation now,
we've already said, is area.

36
00:01:35,860 --> 00:01:39,850
And area, the easiest way to
write area in this form is--

37
00:01:39,850 --> 00:01:42,710
notice that this distance, we
could write it as base times

38
00:01:42,710 --> 00:01:46,302
height or we could write it as
x times y-- so the base here

39
00:01:46,302 --> 00:01:49,250
is x and the height here is y.

40
00:01:49,250 --> 00:01:53,230
So the area of the triangle
is 1/2 base times height.

41
00:01:53,230 --> 00:01:56,350

42
00:01:56,350 --> 00:01:58,490
So the area is 1/2 x times y.

43
00:01:58,490 --> 00:02:01,140
That's the thing we
want to optimize.

44
00:02:01,140 --> 00:02:03,980
The problem is that we know
when we're doing these

45
00:02:03,980 --> 00:02:06,670
optimization problems we want
to take a derivative of area

46
00:02:06,670 --> 00:02:09,400
with respect to a variable,
but right now we have two

47
00:02:09,400 --> 00:02:11,240
variables and so that's
where the constraints

48
00:02:11,240 --> 00:02:12,590
equation comes in.

49
00:02:12,590 --> 00:02:16,470
So now we have to figure out
how we're going to use a

50
00:02:16,470 --> 00:02:17,780
constraint equation here.

51
00:02:17,780 --> 00:02:23,120
The constraint is that it has to
go through this point 8, 4.

52
00:02:23,120 --> 00:02:24,880
So what does our line
have to look like?

53
00:02:24,880 --> 00:02:28,860
Well, our line has to look
like ultimatlely--

54
00:02:28,860 --> 00:02:31,550
let's do, maybe, the
point-slope form--

55
00:02:31,550 --> 00:02:34,240
y is equal, or sorry.

56
00:02:34,240 --> 00:02:35,320
I said point-slope form.

57
00:02:35,320 --> 00:02:39,870
y minus 4 is equal to
m times x minus 8.

58
00:02:39,870 --> 00:02:41,120
Right?

59
00:02:41,120 --> 00:02:43,380

60
00:02:43,380 --> 00:02:45,840
Notice I couldn't
pick what m was.

61
00:02:45,840 --> 00:02:51,460
Because the m completely
determines the line.

62
00:02:51,460 --> 00:02:53,560
So hopefully that make sense,
that you can see that.

63
00:02:53,560 --> 00:02:58,960
Now, in fact, let's look at how
this problem will work.

64
00:02:58,960 --> 00:03:02,150
The m is going to determine this
point and it's going to

65
00:03:02,150 --> 00:03:04,720
determine this point.

66
00:03:04,720 --> 00:03:07,360
If you can't see that, well,
let's look back here.

67
00:03:07,360 --> 00:03:11,145
This point is when y equals 0.

68
00:03:11,145 --> 00:03:12,670
Right?

69
00:03:12,670 --> 00:03:17,940
So I can put in y equals 0 and
I get x in terms of m.

70
00:03:17,940 --> 00:03:19,870
If I come back over here and
look at this point, this is

71
00:03:19,870 --> 00:03:22,900
when x equals 0.

72
00:03:22,900 --> 00:03:26,590
So if I put in 0 for x, I can
find y in terms of m.

73
00:03:26,590 --> 00:03:30,940
So these two values, the x
value and the y value,

74
00:03:30,940 --> 00:03:32,870
completely determine on the
slope of this line.

75
00:03:32,870 --> 00:03:35,790
That hopefully makes sense just
even if you look at the

76
00:03:35,790 --> 00:03:37,610
geometric picture.

77
00:03:37,610 --> 00:03:43,010
When I turn about this point at
8, 4 these values change.

78
00:03:43,010 --> 00:03:45,420
So the x and y values are
completely determined by the

79
00:03:45,420 --> 00:03:46,470
slope of the line.

80
00:03:46,470 --> 00:03:49,000
In fact, the area, then, is
completely determined by the

81
00:03:49,000 --> 00:03:50,520
slope of the line.

82
00:03:50,520 --> 00:03:52,960
So what we're going to do is
we're going to use the

83
00:03:52,960 --> 00:03:56,130
constraint equation to find
x and y values, all in

84
00:03:56,130 --> 00:03:57,890
terms of the slope.

85
00:03:57,890 --> 00:04:00,430
So let's do that.

86
00:04:00,430 --> 00:04:06,710
I said when y is 0, what
do we get for x?

87
00:04:06,710 --> 00:04:14,100
We get negative 4 over m
plus 8 is equal to x.

88
00:04:14,100 --> 00:04:17,100
Let me double check my math so I
don't have to re-shoot this.

89
00:04:17,100 --> 00:04:19,780
When y is 0 I divide by m.

90
00:04:19,780 --> 00:04:23,420
I add 8, I get x.

91
00:04:23,420 --> 00:04:28,780
So that is the x value I'm
interested in down here.

92
00:04:28,780 --> 00:04:31,020
When x is 0--

93
00:04:31,020 --> 00:04:32,360
let's see what I get--

94
00:04:32,360 --> 00:04:35,842
when x is 0 I get negative
8m plus 4 is y.

95
00:04:35,842 --> 00:04:37,092
Right?

96
00:04:37,092 --> 00:04:42,790

97
00:04:42,790 --> 00:04:45,430
x is 0, negative 8m plus 4.

98
00:04:45,430 --> 00:04:47,840
So now what I'm going to do is
plug these two things into the

99
00:04:47,840 --> 00:04:49,090
area equation.

100
00:04:49,090 --> 00:04:51,760

101
00:04:51,760 --> 00:04:55,680
Area is now equal to
1/2 of x times y.

102
00:04:55,680 --> 00:05:04,400
So 1/2 of 8 minus
4 over m times--

103
00:05:04,400 --> 00:05:05,170
you know what I'm going to do?

104
00:05:05,170 --> 00:05:08,050
I'm going to take this 1/2 and
kill off terms in there so I

105
00:05:08,050 --> 00:05:09,910
don't have to worry
about it anymore--

106
00:05:09,910 --> 00:05:12,920
negative 4m plus 2.

107
00:05:12,920 --> 00:05:16,430
So this is x and this
is 1/2 of y.

108
00:05:16,430 --> 00:05:19,630
So just to make it simpler I'm
not carrying through the 1/2--

109
00:05:19,630 --> 00:05:23,620
I'm killing off half of the
things, dividing every

110
00:05:23,620 --> 00:05:25,590
term in y by 2.

111
00:05:25,590 --> 00:05:26,890
And again, what are
we trying to do?

112
00:05:26,890 --> 00:05:28,110
We're trying to optimize.

113
00:05:28,110 --> 00:05:30,700
So now we want to take the
derivative of area with

114
00:05:30,700 --> 00:05:31,950
respect to the slope.

115
00:05:31,950 --> 00:05:35,980

116
00:05:35,980 --> 00:05:38,520
So this is, maybe to simplify
first, let's multiply through.

117
00:05:38,520 --> 00:05:41,530

118
00:05:41,530 --> 00:05:44,260
So this is just a little bit
of algebra really quick.

119
00:05:44,260 --> 00:05:51,130
8 times 4 is 32, so I get
negative 32m plus 16.

120
00:05:51,130 --> 00:05:53,630
And then here, negative times
negative is a positive.

121
00:05:53,630 --> 00:05:54,980
4 times 4 is 16.

122
00:05:54,980 --> 00:05:59,330
m divided by m, I just get 16.

123
00:05:59,330 --> 00:06:02,155
And then here I get
negative 8m.

124
00:06:02,155 --> 00:06:05,750

125
00:06:05,750 --> 00:06:08,490
So I had to do a little bit of
algebra first, but this is

126
00:06:08,490 --> 00:06:10,480
much easier to take a derivative
and not make

127
00:06:10,480 --> 00:06:12,140
mistakes than this one.

128
00:06:12,140 --> 00:06:15,270
Because you'd have a product
rule and then you'd still have

129
00:06:15,270 --> 00:06:16,520
to multiply.

130
00:06:16,520 --> 00:06:19,130
So we might as well multiply
out first.

131
00:06:19,130 --> 00:06:23,080
So now let me just take the
derivative of this.

132
00:06:23,080 --> 00:06:25,690
And again, I'm taking the
derivative with respect to m.

133
00:06:25,690 --> 00:06:27,970
So here I just get
negative 32.

134
00:06:27,970 --> 00:06:28,990
0, 0.

135
00:06:28,990 --> 00:06:30,160
And then what's the
derivative of--

136
00:06:30,160 --> 00:06:32,370
this is a minus 8m--

137
00:06:32,370 --> 00:06:35,550
well, the derivative of 1 over
m, if you remember, is

138
00:06:35,550 --> 00:06:37,670
negative 1 over m squared.

139
00:06:37,670 --> 00:06:40,080
I have another negative here, so
this is going to be plus 8

140
00:06:40,080 --> 00:06:40,710
over m squared.

141
00:06:40,710 --> 00:06:41,960
Right?

142
00:06:41,960 --> 00:06:43,750

143
00:06:43,750 --> 00:06:47,250
Optimizing, we want to set the
derivative equal to 0.

144
00:06:47,250 --> 00:06:52,970
So if I set the derivative equal
to 0 and solve I get 32m

145
00:06:52,970 --> 00:06:58,600
squared equals 8, or m squared
is equal to 8 over 32, which

146
00:06:58,600 --> 00:07:03,580
is 1/4, or m is equal to 1/2.

147
00:07:03,580 --> 00:07:05,600
Or I should say, plus
or minus 1/2.

148
00:07:05,600 --> 00:07:07,580
We need to be aware.

149
00:07:07,580 --> 00:07:10,790
I would run into problems if
I didn't put the minus.

150
00:07:10,790 --> 00:07:14,910
So solving this problem, I see
that-- again, what did I do?

151
00:07:14,910 --> 00:07:18,590
I set area prime equal to 0,
move the 32 over, multiply by

152
00:07:18,590 --> 00:07:21,690
m squared, do some algebra, and
I get m is equal to plus

153
00:07:21,690 --> 00:07:22,710
or minus 1/2.

154
00:07:22,710 --> 00:07:25,830
And now we need to see which of
these things make sense and

155
00:07:25,830 --> 00:07:28,480
then we just need to think about
what happens as m goes

156
00:07:28,480 --> 00:07:30,070
to its extreme values.

157
00:07:30,070 --> 00:07:33,010
So let's come back and look at
the picture and from there we

158
00:07:33,010 --> 00:07:35,720
can probably tell which of
these answers we need.

159
00:07:35,720 --> 00:07:38,650

160
00:07:38,650 --> 00:07:41,480
So it's m equals 1/2 or m equals
minus 1/2 that we want

161
00:07:41,480 --> 00:07:44,680
to know which of these
do we need.

162
00:07:44,680 --> 00:07:47,240
So I'm going to use some
different colored chalk to

163
00:07:47,240 --> 00:07:50,140
draw what's happening here.

164
00:07:50,140 --> 00:07:51,830
Notice the slope of this
line is negative.

165
00:07:51,830 --> 00:07:54,210
Right?

166
00:07:54,210 --> 00:07:57,900
If I were going to do a positive
sloping line, which

167
00:07:57,900 --> 00:08:00,780
would be the case where m is
equal to 1/2, I would get

168
00:08:00,780 --> 00:08:04,410
something that's headed
in this direction.

169
00:08:04,410 --> 00:08:07,270
And notice that that's not going
to make a triangle with

170
00:08:07,270 --> 00:08:09,050
the x- and y-axis.

171
00:08:09,050 --> 00:08:13,070
And so immediately m equals
1/2 isn't even in this

172
00:08:13,070 --> 00:08:15,830
problem, isn't allowed
to work.

173
00:08:15,830 --> 00:08:18,680
OK, now where did
it come from?

174
00:08:18,680 --> 00:08:22,160
It came because somewhere I was
multiplying m by itself,

175
00:08:22,160 --> 00:08:24,830
which maybe isn't actually
in the original part.

176
00:08:24,830 --> 00:08:28,050
I was introducing a new thing
happening, there, so I'm not

177
00:08:28,050 --> 00:08:30,970
going to get into it too much
because we can immediately see

178
00:08:30,970 --> 00:08:33,300
that we don't have to worry
about m equals 1/2.

179
00:08:33,300 --> 00:08:35,670
m equals minus 1/2 looks good.

180
00:08:35,670 --> 00:08:37,540
That's sloping in
this direction.

181
00:08:37,540 --> 00:08:40,860
And in fact, that would give
us a nice triangle.

182
00:08:40,860 --> 00:08:44,970
The extreme values in this case
are obviously when m is

183
00:08:44,970 --> 00:08:49,030
sloping all the way up to being
vertical, or when m is

184
00:08:49,030 --> 00:08:51,240
sloping to being horizontal.

185
00:08:51,240 --> 00:08:53,940
And in both of those cases you
notice that the area is

186
00:08:53,940 --> 00:08:56,170
getting arbitrarily large, it's
headed towards infinity

187
00:08:56,170 --> 00:08:57,320
in both cases.

188
00:08:57,320 --> 00:09:01,240
So I don't need to worry about
looking at the extreme values.

189
00:09:01,240 --> 00:09:03,030
There aren't end points
really in this case.

190
00:09:03,030 --> 00:09:06,420
But the extreme values, they're
both going to positive

191
00:09:06,420 --> 00:09:07,540
infinity, the areas.

192
00:09:07,540 --> 00:09:10,730
Which convinces me even more
that where m is equal to minus

193
00:09:10,730 --> 00:09:13,690
1/2 is going to be a minimum.

194
00:09:13,690 --> 00:09:15,840
You could also take the second
derivative and run the second

195
00:09:15,840 --> 00:09:18,830
derivative test, but even
geometrically, we can see in

196
00:09:18,830 --> 00:09:23,240
the picture that at m equals
negative 1/2 we actually get a

197
00:09:23,240 --> 00:09:25,090
negative sign for the--
or, sorry-- a

198
00:09:25,090 --> 00:09:26,770
minimizer for the area.

199
00:09:26,770 --> 00:09:30,100
And now the question asks
to find the dimensions.

200
00:09:30,100 --> 00:09:32,070
How do I go back and find
the dimensions?

201
00:09:32,070 --> 00:09:34,240
I'm not going to do any more on
this problem, but you can

202
00:09:34,240 --> 00:09:36,350
do it to finish it off.

203
00:09:36,350 --> 00:09:38,950
Finding the dimensions,
I know what m is.

204
00:09:38,950 --> 00:09:41,825
I also know what x is in
terms of m and what y

205
00:09:41,825 --> 00:09:42,870
is in terms of m.

206
00:09:42,870 --> 00:09:46,130
So I just evaluate x at the m
and evaluate y at that m.

207
00:09:46,130 --> 00:09:47,980
That gives me the dimensions
that will

208
00:09:47,980 --> 00:09:49,410
complete the problem.

209
00:09:49,410 --> 00:09:51,380
But I think I'll stop there.

210
00:09:51,380 --> 00:09:51,591