Austin Cholley Zhu MAT 267 ONLINE A Fall 2021
Assignment Section 10.8 due 08/29/2021 at 11:59pm MST
Problem 1. (1 point)
Find the length of the given curve:
r(t) = h−2t,4sint,4costi
where −5≤t≤3.
Solution:
SOLUTION:
r(t) = h−2t,4sint,4costi ⇒ r0(t) = h−2,4cost,−4 sinti
⇒|r0(t)|=q(−2)2+ (4)2cos2t+ (4)2sin2t
=q(−2)2+ (4)2(cos2t+sin2t)
=p(−2)2+ (4)2
=√20
Then L=Z3
−5
r0(t)
dt =Z3
−5
√20 dt =8√20
Answer(s) submitted:
•16sqrt(5)
(correct)
Correct Answers:
•35.7770876399966
Problem 2. (1 point)
Find the arclength of the curve r(t) = D9√2t,e9t,e−9tE, 0 ≤t≤1
Solution:
SOLUTION:
r(t) = D9√2t,e9t,e−9tE⇒r0(t) = D9√2,9e9t,−9e−9tE
⇒|r0(t)|=p2(9)2+ (9)2e18t+ (9)2e−18t
=9√2+e18t+e−18t
=9p(e9t+e−9t)2
=9(e9t+e−9t)
Then
L=R1
0|r0(t)|dt
=R1
09(e9t+e−9t)dt
=e9t−e−9t1
0
= (e9−e−9)−(1−1)
=e9−e−9
Answer(s) submitted:
•eˆ(9)-eˆ(-9)
(correct)
Correct Answers:
•8103.08380416558
Problem 3. (1 point)
Consider the path r(t) = h16t,8t2,8 lntidefined for t>0.
Find the length of the curve between the points (16,8,0)and
(80,200,8ln(5)).
Solution:
SOLUTION:
r(t) = h16t,8t2,8 lnti ⇒ r0(t) = 16,16t,8
t
⇒|r0(t)|=q256 +256t2+82
t2
=8q4+4t2+1
t2
=8q2t+1
t2
=82t+1
t
The two given points correspond to t=1 and t=5, then
L=Z5
1
r0(t)
dt
=Z5
1
82t+1
tdt
=8t2+lnt5
1
=8(25 +ln(5)−1)
Answer(s) submitted:
•
(incorrect)
Correct Answers:
•204.875503299473
1
Problem 4. (1 point)
Consider the helix r(t) = hcos(6t),sin(6t),2ti. Compute, at t=π
6:
A. The unit tangent vector T=h, , i
B. The unit normal vector N=h, , i
C. The unit binormal vector B=h, , i
Solution:
SOLUTION:
r(t) = hcos(6t),sin(6t),2ti
⇒r0(t) = h−6sin(6t),6 cos(6t),2i
⇒|r0(t)|=p(−6sin(6t))2+ (6cos(6t))2+ (2)2=√40.
Then
T(t) = r0(t)
|r0(t)|=1
√40 h−6sin(6t),6cos(6t),2i
and Tπ
6=−6sin (1π)
√40 ,6cos (1π)
√40 ,2
√40
T0(t) = 1
√40 h−(6·6cos(6t)),−6·6 sin(6t),0i
⇒ |T0(t)|=36
√40
Then N(t) = T0(t)
|T0(t)|=h−cos(6t),−sin(6t),0iand
Nπ
6=h−cos(1π),−sin (1π),0i
Bπ
6=Tπ
6×Nπ
6
=D−6sin(1π)
√40 ,6cos(1π)
√40 ,2
√40 E×h−cos(1π),−sin (1π),0i
=D2
√40 sin(1π),−2
√40 cos(1π),6
√40 E
Answer(s) submitted:
•
•
•
•-cos(((6pi )/6))
•-sin(((6pi )/6))
•0
•
•
•
(score 0.333333333333333)
Correct Answers:
•-1.16180196436212E-16
•-0.948683298050514
•0.316227766016838
•1
•-1.22464679914735E-16
•0
•3.87267321454039E-17
•0.316227766016838
•0.948683298050514
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