agrawal (aa87823) – HW15 – clark – (55000)
This print-out should have 8 questions.
Multiple-choice questions may continue on
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before answering.
001
5.
10.0 points
C2 2 Ct,
t ,
12
003
0 ≤ t ≤ 4,
is the arc of the parabola
12y = x2
from (0, 0) to (6, 3), find the coordinates of
the point P on this arc corresponding to t = 2.
3 3 ,
1. P =
2 16
3
2. P = 3 ,
4
3 3
3. P =
,
2 4
3
, 3
4. P =
4
3
5. P =
, 3
16
3 3
,
6. P =
16 2
002
1.
2.
3.
10.0 points
Find a Cartesian equation for the curve
given in parametric form by
x(t) = 2 cos 3t ,
4.
y(t) = 5 sin 3t .
1. 25x2 − 4y 2 = 100
1
x2 y 2
+
=
25
4
100
x2 y 2
1
3.
−
=
4
25
100
2.
4. 4x2 + 25y 2 = 100
x2 y 2
1
−
=
25
4
100
6. 25x2 + 4y 2 = 100
If the constant C is chosen so that the curve
given parametrically by
1
5.
6.
10.0 points
agrawal (aa87823) – HW15 – clark – (55000)
2
y
2.
004
1
1
x
10.0 points
0
0
1
1
Which one of the following could be the
graph of the curve given parametrically by
y
3.
1
1
(x(t), y(t))
when the graphs of x(t) and y(t) are shown in
x
0
0
1
1
1
1
y
4.
1
1
t
0
0
1
1
x(t) :
y(t) :
x
0
0
1
1
y
1.
1
y
1
5.
1
1
x
0
0
1
1
x
0
0
1
1
agrawal (aa87823) – HW15 – clark – (55000)
3
y
6.
1
3. (6 cos t, 2 + 6 sin t),
1
0 ≤ t ≤ 2π
4. ( − 6 cos t, 2 − 6 sin t),
0≤t≤π
5. (6 cos t, 2 − 6 sin t),
0 ≤ t ≤ 2π
6. (6 cos t, 2 − 6 sin t),
0≤t≤π
x
0
0
1
007
1
005
10.0 points
Find parametric equations representing the
line segment joining P (−1, 4) to Q(4, −2) as
(x(t), y(t)) ,
10.0 points
A ladder 13 feet in length slides down a wall
as its bottom is pulled away from the wall as
shown in
y
0 ≤ t ≤ 1,
where
P = (x(0), y(0)) ,
Q = (x(1), y(1))
and x(t), y(t) are linear functions of t.
P
13
1. (−1 + 5t, 4 − 6t)
2.
πt
πt 4 − 5 sin2 , 4 − 6 cos2
2
2
θ
x
3. (4 − 5t, −2 + 6t)
4. (4 − 5t2 , −2 + 6t2 )
5.
πt
πt −1 + 5 sin2 , −2 + 6 cos2
2
2
6. (−1 + 5t2 , 4 − 6t2 )
006
1. (3 cos θ, 10 sin θ)
10.0 points
Find the path (x(t), y(t)) of a particle
that moves once counter-clockwise around the
curve
x2 + (y − 2)2 = 36 ,
starting at (6, 2).
1. (6 cos t, 2 + 6 sin t),
2. ( − 6 cos t, 2 − 6 sin t),
Using the angle θ as parameter, find the
parametric equations for the path followed by
the point P located 3 feet from the top of the
ladder.
2. (3 sin θ, 10 cos θ)
3. (3 sec θ, 10 tan θ)
4. (3 tan θ, 10 sec θ)
5. (10 sin θ, 3 cos θ)
0≤t≤π
0 ≤ t ≤ 2π
6. (10 cos θ, 3 sin θ)
7. (10 tan θ, 3 sec θ)
agrawal (aa87823) – HW15 – clark – (55000)
8. (10 sec θ, 3 tan θ)
008
10.0 points
If a thread is unwound from a stationary
circular spool of radius 7, keeping the thread
taut at all times, then the endpoint P traces
out a curve as shown in
y
Q
P
θ
x
A
called the Involute of the circle.
Using the fact that P Q has length 7θ, find
parametric equations for P using the angle θ
as parameter.
1. 7(cos θ − θ sin θ), 7(sin θ + θ cos θ)
2. 7 sin θ − θ cos θ, 7 cos θ + θ sin θ
3. 7(sin θ − θ cos θ), 7(cos θ + θ sin θ)
4. 7 sin θ + θ cos θ, 7 cos θ − θ sin θ
5. 7 cos θ − θ sin θ, 7 sin θ + θ cos θ
6. 7(sin θ + θ cos θ), 7(cos θ − θ sin θ)
7. 7 cos θ + θ sin θ, 7 sin θ − θ cos θ
8. 7(cos θ + θ sin θ), 7(sin θ − θ cos θ)
4