Mathematics
408
K: Homework
8
Gagan Tara Nanda
09
th
October,
2004
HW
8
:
#
s
8
,
10
,
12
,
20
,
21
,
22
8. I’ve clipped the diagram below from the original solutions, so to have it match my question, change the
length
5
in the diagram to
8
, since the island in my question is
8
miles from the shore.
In this setup,
x
is the distance of the light beam’s endpoint from the left edge of the shore line. We want
to know how fast the beam is travelling along the shore line, so we want to
fi
nd
dx
dt
. From the diagram,
tan
θ
=
x
8
⇒
x
= 8 tan
θ
.
Di
ff
erentiate implicitly with respect to
t
to get
dx
dt
= 8 sec
2
d
θ
dt
.
Observe that
1
revolution is
2
π
radians, so the searchlight turns through
4
·
2
π
= 8
π
radians per minute,
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Gagan Tara Nanda
2
which is
d
θ
dt
. Hence
dx
dt
=
8 sec
2
θ
(8
π
)
=
64
π
cos
2
θ
.
At the particular instant requested, the beam makes an angle of
45
◦
with the shoreline, which is
90
◦
−
θ
.
So
θ
= 90
◦
−
45
◦
= 45
◦
, and we get
dx
dt
=
64
π
cos
2
(45
◦
)
=
64
π
³
1
√
2
´
2
=
128
π
miles/min.
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 Fall '09
 RAdin
 Harshad number, Trigraph, dt dt, DT DT DT, Tara Nanda

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