M408D SPRING 2015 HOMEWORK #6 - cheatham (sc36975) HW06 um...

cheatham (sc36975) – HW06 – um – (53890) 1
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before answering.
001 10.0 points
Determine A so that the curve
y = 6 x + 14
can be written in parametric form as
x ( t ) = t − 3 , y ( t ) = At − 4 .
1. A = 5
2. A = − 4
3. A = 6 correct
4. A = − 5
5. A = 4
6. A = − 6
Determine a Cartesian equation for the
curve given in parametric form by
x ( t ) = 3 e
2 t
, y ( t ) = 4 e
− t
.
002 10.0 points
3. y =
1
2
e
8 /x
4. y =
1
2
e
x/ 8
correct
5. y =
1
4
e
x/ 2
6. y =
1
2
e
x/ 4
Explanation:
We have to eliminate the parameter t from
the equations for x and y . Now from the
equation for x it follows that
t =
1
4
e
x/ 4
.
But then
y =
1
4
e
x/ 4
1 / 2
=
1
2
e
x/ 8
.
003 10.0 points
Determine a Cartesian equation for the
curve given in parametric form by
x ( t ) = 4 ln(4 t ) , y ( t ) =
√
t .
1. y =
1
4
e
x/ 4
2. y =
1
4
e
4 /x
5. xy
2
= 36
6.
x
y 2
= 48
Explanation:
We have to eliminate the parameter t from
the equations for x and y . Now from the
equation for x it follows that
e
t
=
x
3
1 / 2
,

cheatham (sc36975) – HW06 – um – (53890) 2
from which in turn it follows that
y = 4
3
x
1 / 2
.
Consequently,
xy
2
= 48 .
004 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 .
starting at (0 , 3) and ending at (4 , 0).
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
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 .
Consequently, the particle moves clockwise
once around the ellipse
x
2
005 10.0 points
5. Moves once clockwise along the ellipse 6. Moves along the line
x
3
= 1 ,