moseley (cmm3869) – HW07 – Gilbert – (56380)
1
This
print-out
should
have
12
questions.
Multiple-choice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
Find a Cartesian equation for the curve
given in parametric form by
x
(
t
) =
1
4
t
2
,
y
(
t
) =
1
8
t
3
.
1.
x
=
y
2
/
3
correct
2.
x
=
1
2
y
3
/
2
3.
x
=
1
2
y
4
/
3
4.
x
=
1
2
y
2
/
3
5.
x
=
y
3
/
2
6.
x
=
y
4
/
3
Explanation:
We have to eliminate the parameter
t
from
the equations for
x
and
y
.
But from the
equation for
y
, it follows that
t
= 2
y
1
/
3
,
in which case
x
=
1
4
parenleftBig
2
y
1
/
3
parenrightBig
2
=
y
2
/
3
.
002
10.0 points
Describe the motion of a particle with posi-
tion
P
(
x, y
) when
x
= 4 sin
t ,
y
= 3 cos
t
as
t
varies in the interval 0
≤
t
≤
2
π
.
1.
Moves once clockwise along the ellipse
(4
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
2.
Moves once counterclockwise along the
ellipse
x
2
16
+
y
2
9
= 1
,
starting and ending at (0
,
3).
3.
Moves along the line
x
4
+
y
3
= 1
,
starting at (0
,
3) and ending at (4
,
0).
4.
Moves once clockwise along the ellipse
x
2
16
+
y
2
9
= 1
,
starting and ending at (0
,
3).
correct
5.
Moves along the line
x
4
+
y
3
= 1
,
starting at (4
,
0) and ending at (0
,
3).
6.
Moves once counterclockwise along the
ellipse
(4
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
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
16
+
y
2
9
= 1 ;
this is an ellipse centered at the origin.
At
t
= 0, the particle is at (4 sin 0
,
3 cos 0),
i.e.
,
at the point (0
,
3) on the ellipse.
Now as
t
increases from
t
= 0 to
t
=
π/
2,
x
(
t
) increases
from
x
= 0 to
x
= 4, while
y
(
t
) decreases from
y
= 3 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