HW06 – gilbert – (55035)
2
from which in turn it follows that
y
=
2
₃
x
4
₃
1/2
.
Consequently,
xy
2
=
16
.
004
10.0 points
Describe the motion of a particle with posi-
tion
P
(
x, y
) when
x
=
2 sin
t ,
y
=
5 cos
t
as
t
varies in the interval 0 ≤
t
≤ 2
π
.
1.
Moves once clockwise along the ellipse
x
2
+
y
2
=
1
,
4
25
starting and ending at (0
,
5). correct
2.
Moves once counterclockwise along the
ellipse
(2
x
)
2
+ (5
y
)
2
=
1
,
starting and ending at (0
,
5).
3.
Moves along the line
x
2
+
y
5
=
1
,
starting at (0
,
5) and ending at (2
,
0).
4.
Moves along the line
x
2
+
y
5
=
1
,
starting at (2
,
0) and ending at (0
,
5).
5.
Moves once clockwise along the ellipse
(2
x
)
2
+ (5
y
)
2
=
1
,
starting and ending at (0
,
5).
6. Moves once counterclockwise along
the ellipse
x
2
y
2
+
25
=
1
,
4
starting and ending at (0
,
5).
Explanation:
Since
cos
2
t
+ sin
2
t
=
1
for all
t
, the particle travels along the
curve given in Cartesian form by
x
2
+
y
2
=
1 ;
4
25
this is an ellipse centered at the origin. At
t
=
0, the particle is at (2 sin 0
,
5 cos 0),
I.E.,
at the
point (0
,
5) on the ellipse. Now as
t
increases
from
t
= 0 to
t
=
π/
2,
x
(
t
) increases from
x
= 0
to
x
= 2, while
y
(
t
) decreases from
y
= 5 to
y
=
0 ; in particular, the particle
moves from a
point on the positive
y
-axis to a point on the
positive
x
-axis, so it is moving
CLOCKWISE
.
In
the
same
way,
we
see
that
as
t
increases
from
π/
2
to
π
,
the
particle
moves to a point on the negative
y
-axis,
then to a point on the negative
x
-axis as
t
increases from
π
to 3
π/
2, until finally it
returns to its starting point on the positive
y
-axis as
t
increases from 3
π/
2 to 2
π
.