Lecture
12
18.01
Fall
2006
Lecture
12:
Related
Rates
Example
1.
Police
are
30
feet
from
the
side
of
the
road.
Their
radar
sees
your
car
approaching
at
80
feet
per
second
when
your
car
is
50
feet
away
from
the
radar
gun.
The
speed
limit
is
65
miles
per
hour
(which
translates
to
95
feet
per
second).
Are
you
speeding?
First,
draw
a
diagram
of
the
setup
(as
in
Fig.
1):
Road
Car
Police
30
D=50
x
Figure
1:
Illustration
of
example
1:
triangle
with
the
police,
the
car,
the
road,
D
and
x
labelled.
Next,
give
the
variables
names.
The
important
thing
to
figure
out
is
which
variables
are
changing.
dD
At
D
=
50,
x
=
40.
(We
know
this
because
it’s
a
345
right
triangle.)
In
addition,
=
D
=
dt

80.
D
is
negative
because
the
car
is
moving
in
the

x
direction.
Don’t
plug
in
the
value
for
D
yet!
D
is
changing,
and
it
depends
on
x
.
The
Pythagorean
theorem
says
30
2
+
x
2
=
D
2
Di
ff
erentiate
this
equation
with
respect
to
time
(implicit
di
ff
erentiation:
d
(
2
)
2
DD
30
2
+
x
=
D
2
=
2
xx
= 2
DD
=
x
=
dt
⇒
⇒
2
x
Now,
plug
in
the
instantaneous
numerical
values:
50
feet
x
=
40
(

80)
=

100
s
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 Fall '08
 Staff
 Pythagorean Theorem, triangle, 5 feet, 4 feet, parabolic mirror problem

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