1
00:00:00,000 --> 00:00:00,510


2
00:00:00,510 --> 00:00:03,780
Let's do some word problems that are essentially dealing

3
00:00:03,779 --> 00:00:05,969
with slope of a line.

4
00:00:05,969 --> 00:00:08,250
You might see these being referred to as direct

5
00:00:08,250 --> 00:00:11,279
variation models, because we're going to model what's

6
00:00:11,279 --> 00:00:13,589
being described in this problem.

7
00:00:13,589 --> 00:00:15,044
We're going to graph it out and then hopefully we're going

8
00:00:15,044 --> 00:00:17,620
to be able actually answer their question.

9
00:00:17,620 --> 00:00:18,540
Let's see what they're asking.

10
00:00:18,539 --> 00:00:23,750
The current standard for low-flow showerheads is 2.5

11
00:00:23,750 --> 00:00:25,000
gallons per minute.

12
00:00:25,000 --> 00:00:32,189


13
00:00:32,189 --> 00:00:35,390
Calculate how long it would take to fill a 30 gallon

14
00:00:35,390 --> 00:00:39,179
bathtub using such a showerhead

15
00:00:39,179 --> 00:00:41,340
to supply the water.

16
00:00:41,340 --> 00:00:44,010
What we can do here is we can set up a direct variation

17
00:00:44,009 --> 00:00:46,309
model, which sounds very fancy, but it just says, OK,

18
00:00:46,310 --> 00:00:50,690
let's set up a little equation that describes how many

19
00:00:50,689 --> 00:00:53,609
gallons we would have filled-- or how many gallons we would

20
00:00:53,609 --> 00:00:58,979
have used after a certain number of minutes.

21
00:00:58,979 --> 00:01:05,599
So let's say that we have gallons is equal to the rate

22
00:01:05,599 --> 00:01:06,780
at which we fill the gallons.

23
00:01:06,780 --> 00:01:17,460
So it's going to be 2.5 gallons per minute times the

24
00:01:17,459 --> 00:01:18,709
number of minutes.

25
00:01:18,709 --> 00:01:21,059


26
00:01:21,060 --> 00:01:24,379
I'll say m for minutes and g for gallons.

27
00:01:24,379 --> 00:01:27,859
We have just set up our direct variation model.

28
00:01:27,859 --> 00:01:28,780
Nothing fancier than that.

29
00:01:28,780 --> 00:01:31,359
We now have an equation that describes, you give me the

30
00:01:31,359 --> 00:01:34,429
number of minutes, I'll multiply that times 2.5

31
00:01:34,430 --> 00:01:37,550
because that's how quickly we're filling the bathtub.

32
00:01:37,549 --> 00:01:41,459
So after 1 minute, 1 times 2.5, we have 2.5 gallons.

33
00:01:41,459 --> 00:01:45,219
After 2 minutes, we have 2 times 2.5, we have 5 gallons.

34
00:01:45,219 --> 00:01:46,329
So this is our model.

35
00:01:46,329 --> 00:01:48,349
And this is also a line.

36
00:01:48,349 --> 00:01:53,129
Remember the form of a line is y is equal to mx plus b.

37
00:01:53,129 --> 00:01:54,509
Here we have no b.

38
00:01:54,510 --> 00:01:55,320
The b is gone.

39
00:01:55,319 --> 00:01:57,199
We just have an m times an x.

40
00:01:57,200 --> 00:02:00,250
The x now we're calling minutes and the y we're now

41
00:02:00,250 --> 00:02:03,760
calling gallons and the slope is now 2.5.

42
00:02:03,760 --> 00:02:07,670
Let's plot this before I even answer their question.

43
00:02:07,670 --> 00:02:10,689
Instead of calling this the x-axis-- remember x, the

44
00:02:10,689 --> 00:02:12,729
independent variable, is now the minutes

45
00:02:12,729 --> 00:02:13,849
that we let it flow.

46
00:02:13,849 --> 00:02:16,469
So that's the m-axis for minutes.

47
00:02:16,469 --> 00:02:19,069
And the vertical axis, instead of calling it the y-axis, I'm

48
00:02:19,069 --> 00:02:21,259
going to call it the g-axis for the number of gallons

49
00:02:21,259 --> 00:02:22,699
we've filled.

50
00:02:22,699 --> 00:02:24,865
I'm only going to deal in the positive quadrant assuming we

51
00:02:24,866 --> 00:02:26,710
can only have positive minutes.

52
00:02:26,710 --> 00:02:27,620
So what's going on here?

53
00:02:27,620 --> 00:02:29,819
We have a slope of 2.5.

54
00:02:29,819 --> 00:02:32,900
We could also write this as gallons is equal to-- 2.5 is

55
00:02:32,900 --> 00:02:42,930
the same thing as 5/2 gallons per minute times the minutes.

56
00:02:42,930 --> 00:02:45,030
So now we know our slope is 5/2.

57
00:02:45,030 --> 00:02:48,280
I could have used 2.5 as well, but I like 5/2.

58
00:02:48,280 --> 00:02:50,219
Our y-intercept we already know is 0.

59
00:02:50,219 --> 00:02:51,699
There's no y-intercept here.

60
00:02:51,699 --> 00:02:54,049
You could do this as plus 0.

61
00:02:54,050 --> 00:02:55,990
So you start over here at the origin.

62
00:02:55,990 --> 00:02:57,379
That's our y-intercept.

63
00:02:57,379 --> 00:03:00,280
And for every 2 that we go to the right, we move up 5.

64
00:03:00,280 --> 00:03:02,240
So we move-- change in x is 2.

65
00:03:02,240 --> 00:03:05,120
One, two, three, four, five.

66
00:03:05,120 --> 00:03:06,180
Change in x is 2.

67
00:03:06,180 --> 00:03:09,159
One, two, three, four, five.

68
00:03:09,159 --> 00:03:11,549
If change in x is negative 2, then change in y, it'll be

69
00:03:11,550 --> 00:03:12,630
negative 5.

70
00:03:12,629 --> 00:03:13,539
Negative 2.

71
00:03:13,539 --> 00:03:16,919
One, two, three, four, five.

72
00:03:16,919 --> 00:03:18,299
And so on and so forth.

73
00:03:18,300 --> 00:03:19,300
We'll eventually get down here.

74
00:03:19,300 --> 00:03:23,755
So our line, our model if we graph it, looks like this.

75
00:03:23,754 --> 00:03:26,799
I'm trying my best to go through all of the points.

76
00:03:26,800 --> 00:03:28,360
Actually, I just said I would only do it

77
00:03:28,360 --> 00:03:30,050
in the first quadrant.

78
00:03:30,050 --> 00:03:32,810
It really doesn't make sense in this quadrant right here,

79
00:03:32,810 --> 00:03:34,900
because you can't have negative minutes.

80
00:03:34,900 --> 00:03:37,370
Or you shouldn't have negative minutes.

81
00:03:37,370 --> 00:03:40,240
So we should only be dealing with it up here.

82
00:03:40,240 --> 00:03:43,560
Now they're asking how long will it take to fill a

83
00:03:43,560 --> 00:03:45,539
30-gallon bathtub?

84
00:03:45,539 --> 00:03:48,539
Now, unfortunately, my graph right here does not go all the

85
00:03:48,539 --> 00:03:49,509
way up to 30 gallons.

86
00:03:49,509 --> 00:03:52,799
If it did-- this is 10 gallons right up here.

87
00:03:52,800 --> 00:03:55,390
That's 10 gallons if I went 3 times higher, I could just

88
00:03:55,389 --> 00:03:56,369
read the graph.

89
00:03:56,370 --> 00:03:59,780
But we could also solve for it algebraically right here.

90
00:03:59,780 --> 00:04:01,150
How many minutes does it take?

91
00:04:01,150 --> 00:04:03,870
Well, let's just make our gallons equal to 30.

92
00:04:03,870 --> 00:04:09,450
So you have 30 gallons is equal to, I'll put it in the

93
00:04:09,449 --> 00:04:18,278
same color, 2.5 gallons per minute times minutes.

94
00:04:18,278 --> 00:04:21,509


95
00:04:21,509 --> 00:04:24,800
Now, all we have to do to solve for minutes is divide

96
00:04:24,800 --> 00:04:28,400
both sides by 2.5 gallons per minute.

97
00:04:28,399 --> 00:04:34,949
Divide both sides by 2.5 gallons per minute.

98
00:04:34,949 --> 00:04:41,039


99
00:04:41,040 --> 00:04:43,080
I'm doing the units to show you that the units actually

100
00:04:43,079 --> 00:04:43,969
all work out in the end.

101
00:04:43,970 --> 00:04:45,170
So this will cancel out.

102
00:04:45,170 --> 00:04:47,280
It will just become a 1.

103
00:04:47,279 --> 00:04:52,449
And so the left-hand side-- or we could say m-- is going to

104
00:04:52,449 --> 00:04:57,099
be equal to 30 divided by 2.5.

105
00:04:57,100 --> 00:04:58,770
We have this gallons in the numerator.

106
00:04:58,769 --> 00:05:01,889


107
00:05:01,889 --> 00:05:03,579
I want to show you, you can deal with the units just like

108
00:05:03,579 --> 00:05:06,180
you would deal with actual numbers.

109
00:05:06,180 --> 00:05:08,569
And if I have gallons per minute in the denominator, if

110
00:05:08,569 --> 00:05:10,730
I divide by this fraction, that's the same thing is

111
00:05:10,730 --> 00:05:14,840
multiplying by its inverse.

112
00:05:14,839 --> 00:05:17,779
That's the same thing as multiplying

113
00:05:17,779 --> 00:05:23,229
by minutes per gallon.

114
00:05:23,230 --> 00:05:24,020
Right?

115
00:05:24,019 --> 00:05:24,959
These units were in the denominator.

116
00:05:24,959 --> 00:05:27,849
When I put them in the numerator, I flip them.

117
00:05:27,850 --> 00:05:31,070
So gallons in the numerator, gallons in the denominator.

118
00:05:31,069 --> 00:05:32,199
They cancel out.

119
00:05:32,199 --> 00:05:37,279
So I'm left with 30 divided by 2.5 minutes.

120
00:05:37,279 --> 00:05:44,899
And what is 30 divided by 2.5?

121
00:05:44,899 --> 00:05:47,109
Is equal to 12.

122
00:05:47,110 --> 00:05:52,020
So it'll take us 12 minutes to fill up a 30 gallon bathtub.

123
00:05:52,019 --> 00:05:53,269
We have our minutes right there.

124
00:05:53,269 --> 00:05:54,879
12 minutes.

125
00:05:54,879 --> 00:05:57,850
Let's do one more.

126
00:05:57,850 --> 00:06:00,939
Amen is-- or maybe A-man, I don't know the best way to

127
00:06:00,939 --> 00:06:03,750
pronounce that name-- is using a hose-- let me scroll over a

128
00:06:03,750 --> 00:06:06,879
little bit-- is using a hose to fill his new swimming pool

129
00:06:06,879 --> 00:06:08,069
for the first time.

130
00:06:08,069 --> 00:06:10,849
He starts the hose at 10 PM-- let me write this down.

131
00:06:10,850 --> 00:06:18,420
He starts at 10 PM-- so this is the start time-- and leaves

132
00:06:18,420 --> 00:06:19,540
it running all night.

133
00:06:19,540 --> 00:06:22,150
At 6 AM, he measures the depth and calculates

134
00:06:22,149 --> 00:06:24,519
the pool is 4/7 full.

135
00:06:24,519 --> 00:06:28,299
So when he starts at 10 PM-- so this is time and this is

136
00:06:28,300 --> 00:06:30,470
how full the pool is.

137
00:06:30,470 --> 00:06:32,680
So, obviously when he starts it, the pool is empty.

138
00:06:32,680 --> 00:06:33,759
It's a new swimming pool.

139
00:06:33,759 --> 00:06:34,240
They tell us.

140
00:06:34,240 --> 00:06:37,530
So the pool is 0 full.

141
00:06:37,529 --> 00:06:39,299
It's 0 whatever.

142
00:06:39,300 --> 00:06:42,500
It has no water in it whatsoever.

143
00:06:42,500 --> 00:06:49,790
Then at 6 AM, he measures the depth and calculates the pool

144
00:06:49,790 --> 00:06:51,660
is 4/7 full.

145
00:06:51,660 --> 00:06:54,920
So here it is 4/7 full.

146
00:06:54,920 --> 00:06:58,160
At what time will his new pool be full?

147
00:06:58,160 --> 00:06:59,530
So we want to know when it's full.

148
00:06:59,529 --> 00:07:01,109
When it is 1/1 full?

149
00:07:01,110 --> 00:07:02,500
Where it's 7/7 full.

150
00:07:02,500 --> 00:07:04,420
At what time?

151
00:07:04,420 --> 00:07:07,290
So to do this, we have to set up a similar model that we did

152
00:07:07,290 --> 00:07:08,500
the last time.

153
00:07:08,500 --> 00:07:13,769
We could say, the fullness of the pool is equal to some

154
00:07:13,769 --> 00:07:17,169
constant times the amount of time that's passed by.

155
00:07:17,170 --> 00:07:21,319


156
00:07:21,319 --> 00:07:24,730
We know when time is equal to 0-- let me put this this way.

157
00:07:24,730 --> 00:07:25,910
This is time.

158
00:07:25,910 --> 00:07:26,490
Let me write it here.

159
00:07:26,490 --> 00:07:27,980
This is time.

160
00:07:27,980 --> 00:07:29,870
This is time, 0.

161
00:07:29,870 --> 00:07:31,920
What is this in hours?

162
00:07:31,920 --> 00:07:34,100
This is 8 hours later, right?

163
00:07:34,100 --> 00:07:35,640
This is time 8.

164
00:07:35,639 --> 00:07:37,149
We don't know what this is.

165
00:07:37,149 --> 00:07:38,659
This is time something else.

166
00:07:38,660 --> 00:07:44,320
So when time is 0 at 10 PM, 0 times k, we have 0 fullness.

167
00:07:44,319 --> 00:07:46,240
We are not full at all.

168
00:07:46,240 --> 00:07:53,180
When time is equal to 8, we have k times 8. k is the rate

169
00:07:53,180 --> 00:07:55,209
at which we're filling the pool.

170
00:07:55,209 --> 00:07:56,060
k times 8.

171
00:07:56,060 --> 00:07:58,600
We're at 4/7 fullness.

172
00:07:58,600 --> 00:08:01,490
So now we can actually figure out what k is.

173
00:08:01,490 --> 00:08:04,480
We can figure out what our proportionality constant is

174
00:08:04,480 --> 00:08:06,460
for our direct variation model.

175
00:08:06,459 --> 00:08:10,789
Sounds very fancy, but all we're saying is, look, this

176
00:08:10,790 --> 00:08:13,120
pool-filling business can be modeled by an

177
00:08:13,120 --> 00:08:14,550
equation like this.

178
00:08:14,550 --> 00:08:17,199
The amount that we're full is directly proportional to the

179
00:08:17,199 --> 00:08:19,789
amount of time that we let the hose run.

180
00:08:19,790 --> 00:08:21,490
And this is the proportionality constant.

181
00:08:21,490 --> 00:08:24,280
We don't know how quickly it fills, but now we can figure

182
00:08:24,279 --> 00:08:25,799
out how quickly it fills.

183
00:08:25,800 --> 00:08:31,889
Because we know after 8 hours it is 4/7 full.

184
00:08:31,889 --> 00:08:35,168
So to solve for k, you divide both sides by 8 hours.

185
00:08:35,168 --> 00:08:45,370
So we get k is equal to 4/7 full divided by 8 hours, which

186
00:08:45,370 --> 00:08:53,149
is the same thing as 4/7 times 1/8 full per hour.

187
00:08:53,149 --> 00:08:56,139
So if we figure this out, let's see.

188
00:08:56,139 --> 00:08:56,980
Divide by 4.

189
00:08:56,980 --> 00:08:58,360
Divide by 4.

190
00:08:58,360 --> 00:09:02,050
So we get 1/14 fullness.

191
00:09:02,049 --> 00:09:04,419
It's kind of a weird unit-- full per hour.

192
00:09:04,419 --> 00:09:08,689
Or you could say, we fill 1/14 of the pool per hour.

193
00:09:08,690 --> 00:09:10,920
So k is 1/14.

194
00:09:10,919 --> 00:09:15,240
So here our equation-- let me write over here.

195
00:09:15,240 --> 00:09:23,299
The fullness of the pool is equal to 1/14 times the time.

196
00:09:23,299 --> 00:09:25,129
So the question that we have to answer is when

197
00:09:25,129 --> 00:09:28,240
does this equal 1?

198
00:09:28,240 --> 00:09:29,909
At what time?

199
00:09:29,909 --> 00:09:31,730
So let's set up the equation.

200
00:09:31,730 --> 00:09:32,720
So we have 1.

201
00:09:32,720 --> 00:09:34,035
That means we're completely full.

202
00:09:34,034 --> 00:09:38,029
It is equal to 1/14 times time.

203
00:09:38,029 --> 00:09:44,519
If we multiply both sides of this equation times 14, the

204
00:09:44,519 --> 00:09:46,419
1/14 and the 14 cancel out.

205
00:09:46,419 --> 00:09:50,949
And we're left with t is equal to 14.

206
00:09:50,950 --> 00:09:54,240
So the pool will fill after 14 hours.

207
00:09:54,240 --> 00:09:56,490
Remember everything we were dealing with was hours.

208
00:09:56,490 --> 00:09:59,870
If we start at 10 PM-- that was time 0-- what time are we

209
00:09:59,870 --> 00:10:01,860
at 14 hours?

210
00:10:01,860 --> 00:10:06,240
So 10 PM in one day-- if you go to 10 AM the next day,

211
00:10:06,240 --> 00:10:08,279
that's 12 hours.

212
00:10:08,279 --> 00:10:09,009
That's how I think.

213
00:10:09,009 --> 00:10:14,600
We need to go 2 more hours to get to 14 hours.

214
00:10:14,600 --> 00:10:18,670
That's noon the next day.

215
00:10:18,669 --> 00:10:22,490
At noon the day after he starts filling is when the

216
00:10:22,490 --> 00:10:24,190
pool will be full.

217
00:10:24,190 --> 00:10:25,260
We can graph this.

218
00:10:25,259 --> 00:10:26,289
I had this graph paper here.

219
00:10:26,289 --> 00:10:28,819
Let's actually graph everything I'm talking about.

220
00:10:28,820 --> 00:10:32,470
The equation-- I wrote it over here-- fullness is equal to

221
00:10:32,470 --> 00:10:35,620
1/14 times t.

222
00:10:35,620 --> 00:10:37,860
Let's assume that each of these notches is two.

223
00:10:37,860 --> 00:10:44,690
That is two, four, six, eight, ten, twelve, fourteen.

224
00:10:44,690 --> 00:10:48,990
So this tells us that as we run 14, we rise 1.

225
00:10:48,990 --> 00:10:54,360
So if x-change is 14, y-change is positive 1.

226
00:10:54,360 --> 00:10:56,360
And I'll make these units 1, 2.

227
00:10:56,360 --> 00:10:58,600
So the scale isn't exactly perfect.

228
00:10:58,600 --> 00:11:01,370
I'm distorting the graph a little bit, but that is 1.

229
00:11:01,370 --> 00:11:04,480
So the graph will look something like that.

230
00:11:04,480 --> 00:11:05,340
Just like that.

231
00:11:05,340 --> 00:11:07,690
It has a slope of 1/14.

232
00:11:07,690 --> 00:11:10,210
Anyway, hopefully you found that useful.

233
00:11:10,210 --> 00:11:10,533