NEWrelated_rates

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PROBLEM


1


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PROBLEM


1
 A
screen
saver
displays
the
outline
of
a
3
cm
by
2
cm
 rectangle
and
then
expands
the
rectangle
in
such
a
way
 that
the
2
cm
side
is
expanding
at
the
rate
of
4
cm/sec

 and
the
propor=ons
of
the
rectangle
never
change.

 How
fast
is
the
area
of
the
rectangle
increasing
when
its
 dimensions
are
12
cm
by
8
cm?
 MATHEMATICAL
MODEL
OF
THE
PROBLEM
 A
screen
saver
displays
the
outline
of
a
3
cm
by
2
cm
rectangle
and
then
expands

 the
rectangle
in
such
a
way
that
the
2
cm
side
is
expanding
at
the
rate
of
4
cm/sec

 and
the
propor=ons
of
the
rectangle
never
change.

 How
fast
is
the
area
of
the
rectangle
increasing
when
its
dimensions
are

 12
cm
by
8
cm?
 Diagram:
 A
 x
 y
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:
x,

y

and


A
 














dimensions






area

 Rela%ons
among
 variables:
 • A=xy
 • x=(3/2)y
 Known:


dy/dt
=
4
cm/sec
 Unknown:
dA/dt

when
x=12
and
y=8
 Diagram:
 A
 x
 y
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:
x,

y

and


A
 














dimensions






area

 Rela%ons
among
 variables:
 • A=xy
 • x=(3/2)y
 Known:


dy/dt
=
4
cm/sec
 Unknown:
dA/dt

when
x=12
and
y=8
 Rates:
 • dx/dt
 Step
1:
List
the
rates
 • dy/dt
 • dA/dt
 Rela=on

 Chain
rule
 Rela=on

 among

 among

 variables
 rates
 Step
2:
List
the
rela=ons
 













among
the
rates
 Variables:
 • 
independent:
t
 • 
dependent:
x,

y

and


A
 Rela%ons
among
 variables:
 • A=xy
 • x=(3/2)y
 Rates:
 • dx/dt
 • dy/dt
 • dA/dt
 Known:


dy/dt
=
4
cm/sec
 Unknown:
dA/dt


 



















when
x=12
and
y=8
 Rela=on

 Chain
rule
 Rela=on

 among

 among

 variables
 rates
 unknown
 dA A = xy ⇒ dt 3 dx x= y⇒ 2 dt dx dy = y+x dt dt ⇒ 3 dy = 2 dt known
 dy dA 3 dy dx 3 = y+x = y + x dt dt 2 dt dt 2 Plug
in:

 x=12,
y=3,
dy/dt=4
 ⇒ dA 3 = 8 + 12 4 = 96 cm 2 / sec dt 2 






PROBLEM


2
 An
FBI
agent
with
a
powerful
spyglass
is
located
in
a
boat
 anchored
400
meters
offshore.
A
gangster
under
 surveillance
is
walking
along
the
shore.

 Assuming
the
shoreline
is
straight
and
that
the
gangster
is
 walking
at
the
rate
of
2
km/hr,
how
fast
must
the
FBI
agent
 rotate
the
spyglass
to
track
the
gangster
when
the
gangster
 is
1
km
from
the
point
on
the
shore
nearest
to
the
boat.
 MATHEMATICAL
MODEL
OF
THE
PROBLEM
 An
FBI
agent
with
a
powerful
spyglass
is
located
in
a
boat
anchored
400
meters
 offshore.
A
gangster
under
surveillance
is
walking
along
the
shore.
Assuming
the
 shoreline
is
straight
and
that
the
gangster
is
walking
at
the
rate
of
2
km/hr,
how
 fast
must
the
FBI
agent
rotate
the
spyglass
to
track
the
gangster
when
the
gangster
 is
1
km
from
the
point
on
the
shore
nearest
to
the
boat.
 Diagram:
 m
 gangster
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:




x and θ
 Rela%ons
among
 variables:
 • x=.4
tanθ
 FBI
agent
 Known:


dx/dt
=
2
km/hr
 Unknown:
dθ/dt

when
x=1
km
 





















distance






angle

 Diagram:
 m
 gangster
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:




x and θ
 





















distance






angle

 Rela%ons
among
 variables:
 • x=(.4)
tanθ
 FBI
agent
 Known:


dx/dt
=
2
km/hr
 Unknown:
dθ/dt

when
x=1
km
 Rates:
 • dx/dt
 Step
1:
List
the
rates
 • dθ/dt
 Step
2:
List
the
rela=ons
 













among
the
rates
 Rela=on

 Chain
rule
 Rela=on

 among

 among

 variables
 rates
 Variables:
 • independent:
t
 • 
dependent:
x
and

θ
 Rela%ons
among
 variables:
 • x=(.4)
tanθ
 Rates:
 • dx/dt
 • dθ/dt
 Known:


dx/dt
=
2
km/hr
 Unknown:
dθ/dt

when
x=1
km
 known
 unknown
 dx dθ dθ 1 dx cos2 θ dx x = (.4 ) tan θ ⇒ = (.4 ) sec 2 θ ⇒ = = 2 dt dt dt (.4 ) sec θ dt (.4 ) dt We
know:
 dx/dt=2km/hr,
x=1km
 Need:
cosθ
 C
 cosθ = AC (.4 ) =2 BC 1 + (.4 ) 2 2 2 m
 A
 =1km
 B
 dA cos θ dx (.4 ) = = 2 = .7 rad / hr 2 2 dt (.4 ) dt (.4 )[1 + (.4 ) ] € BC = 12 + (.4 ) 2 km 






PROBLEM


3
 A
ladder
10
Z
long
rests
against
a
ver=cal
wall.
If
the
 bo[om
of
the
ladder
slides
away
from
the
wall
at
a
rate
of
 1
Z/s,
how
fast
is
the
top
of
the
ladder
sliding
down
the
 wall
when
the
bo[om
of
the
ladder
is
6
Z
from
the
wall?
 MATHEMATICAL
MODEL
OF
THE
PROBLEM
 A
ladder
10
Z
long
rests
against
a
ver=cal
wall.
If
the
 bo[om
of
the
ladder
slides
away
from
the
wall
at
a
rate
of
1
Z/s,
how
fast
is
the
 top
of
the
ladder
sliding
down
the
wall
when
the
bo[om
of
the
ladder
is
6
Z
from
 the
wall?
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:




x and y
 





















distance






height
 





















from
wall


 Rela%ons
among
 variables:
 • x2+y2=102
 Known:


dx/dt
=
1
Z/s
 Unknown:
dy/dt

when
x=6
Z
 Variables:
 































=me
 • 
independent:
t
 • 
dependent:




x and y
 





















distance






height
 





















from
wall


 Known:


dx/dt
=
1
Z/s
 Unknown:
dy/dt

when
x=6
Z
 Rates:
 • dx/dt
 Step
1:
List
the
rates
 • dy/dt
 Rela%ons
among
 variables:
 • x2+y2=102
 Step
2:
List
the
rela=ons
 













among
the
rates
 Rela=on

 Chain
rule
 Rela=on

 among

 among

 variables
 rates
 Variables:
 • independent:
t
 • 
dependent:
x
and

y
 Rela%ons
among
 variables:
 • x2+y2=102
 Rates:
 • dx/dt
 • dy/dt
 Known:


dx/dt
=
1
Z/s
 Unknown:
dy/dt

when
x=6
Z
 known
 unknown
 x 2 + y 2 = 100 ⇒ 2 x dx dy dy − x dx + 2y =0⇒ = dt dt dt y dt We
know:
 dx/dt=1
Z/s,
x=
6
Z
 y = 100 − 36 = 64 = 8 Need:
y
 € dy − x dx 6 3 = = − 1 = − ft / s dt y dt 8 4 y = 100 − 36 = 8 y=?
 6
 € ...
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This note was uploaded on 12/07/2010 for the course MATH MATH 2A taught by Professor Alessandrapantano during the Fall '10 term at UC Irvine.

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