MAT_267_Section 12.7.pdf - Austin Cholley Assignment...

Austin Cholley Zhu MAT 267 ONLINE A Fall 2021
Assignment Section 12.7 due 09/26/2021 at 11:59pm MST
Problem 1.
(1 point)
What are the rectangular coordinates of the point whose spherical
coordinates are
(
3 ,
7 π
6
,
2 π
3
)
?
x =
y =
z =
Problem 2.
(1 point)
What are the spherical coordinates of the point whose rectangular
coordinates are
( 2 , 5 , - 1 ) ?
ρ =
θ =
φ =
1

Problem 3.
(1 point)
Express the point given in Cartesian coordinates in spherical co-
ordinates ( ρ , θ , φ ) . Note: you really only have to do the work for
one, if you use a little geometry and your knowledge of the trig
functions.
A)
7
4
√
2 ,
7
4
√
2 , -
7
2
√
3 =
B) -
7
4
√
2 ,
7
4
√
2 , -
7
2
√
3 =
C)
7
4
√
2 , -
7
4
√
2 , +
7
2
√
3 =
D) -
7
4
√
2 , -
7
4
√
2 , +
7
2
√
3 =
Solution:
SOLUTION
A) ρ
2
=
p
x 2 + y 2 + z 2 =
r
7
4
√
2
2
+
7
4
√
2
2
+
(
-
7
2
√
3
)
2
=
√
7 2 = 7
The point
7
4
√
2 ,
7
4
√
2 is in the first quadrant of the xy -plane, so
θ = arctan
(
y
x
)
= arctan
7
4
√
2 /
7
4
√
2 = arctan ( 1 ) =
π
4
cos ( φ ) =
z
ρ
=
-
7
2
√
3
7
= -
1
2
√
3 ⇒ φ =
5 π
6
Thus spherical coordinates are 7 ,
π
4
,
5 π
6
.
B) ρ
2
=
p
x 2 + y 2 + z 2 =
r
-
7
4
√
2
2
+
7
4
√
2
2
+
(
-
7
2
√
3
)
2
=
√
7 2 = 7
The point -
7
4
√
2 ,
7
4
√
2 is in the second quadrant of the xy -
plane, so θ = arctan
(
y
x
)
+ π = arctan
7
4
√
2 / -
7
4
√
2 + π =
arctan ( - 1 )+ π = -
π
4
+ π =
3 π
4
cos ( φ ) =
z
ρ
=
-
7
2
√
3
7
= -
1
2
√
3 ⇒ φ =
5 π
6
Thus spherical coordinates are 7 ,
3 π
4
,
5 π
6
.
C) ρ
2
=
p
x 2 + y 2 + z 2 =
r
7
4
√
2
2
+ -
7
4
√
2
2
+
(
+
7
2
√
3
)
2
=
√
7 2 = 7
The point
7
4
√
2 , -
7
4
√
2 is in the fourth quadrant of the xy -
plane, so θ = arctan
(
y
x
)
+ 2 π = arctan -
7
4
√
2 /
7
4
√
2 + 2 π =
arctan ( - 1 )+ 2 π = -
π
4
+ 2 π =
7 π
4
cos ( φ ) =
z
ρ
=
+
7
2
√
3
7
= +
1
2
√
3 ⇒ φ =
π
6
Thus spherical coordinates are 7 ,
7 π
4
,
π
6
.
D) ρ
2
=
p
x 2 + y 2 + z 2 =
r
-
7
4
√
2
2
+ -
7
4
√
2
2
+
(
-
7
2
√
3
)
2
=
√
7
2
= 7
The point -
7
4
√
2 , -
7
4
√
2 is in the third quadrant of the xy -
plane, so θ = arctan
(
y
x
)
+ π = arctan -
7
4
√
2 / -
7
4
√
2 + π =
arctan ( 1 )+ π =
π
4
+ π =
5 π
4
cos ( φ ) =
z
ρ
=
+
7
2
√
3
7
= +
1
2
√
3 ⇒ φ =
π
6
Thus spherical coordinates are 7 ,
5 π
4
,
π
6
.
Answer(s) submitted:
• (7,arctan(1),((7pi )/4))
• (7,)
• (7,)
• (7,)
(incorrect)
Correct Answers:
• (7,0.785398,2.61799)
• (7,2.35619,2.61799)
• (7,5.49779,0.523599)
• (7,3.92699,0.523599)
Problem 4.
(1 point)
What are the cylindrical coordinates of the point whose spherical
coordinates are
( ρ , θ , φ ) = ( 5 , 1 ,
2 π
3
) ?
r =
θ =
z =
Solution:
SOLUTION
r = ρ sin ( φ ) = 5sin
(
2 π
3
)
= 5
√
3
2
= ( 5 / 2 )
√
3
θ = 1
z = ρ cos ( φ ) = 5cos
(
2 π
3
)
= 5
(
-
1
2
)
= - 5 / 2
Thus cylindrical coordinates are
(
( 5 / 2 )
√
3 , 1 , - 5 / 2
)
Answer(s) submitted:
• 4.3301
• 1
• -2.5
(correct)
Correct Answers:
• 4.33013
• 1
• -2.5
2