MAT267_10.1_Jones
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School
Arizona State University *
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Course
267
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
6
Uploaded by CorporalOysterPerson1054
Kendall Painley
Jones
MAT
267
ONLINE
B
Spring
2024
Assignment Section
10.1 due 03/17/2024 at 11:59pm MST
Problem 1.
(1 point)
What are the projections of the point
(
−
2
,
4
,
−
1
)
on the coordi-
nate planes?
On the xy-plane: (
,
,
)
On the yz-plane: (
,
,
)
On the xz-plane: (
,
,
)
Solution:
SOLUTION:
The projection of
(
−
2
,
4
,
−
1
)
onto the
xy
-plane is
(
−
2
,
4
,
0
)
.
The projection of
(
−
2
,
4
,
−
1
)
onto the
yz
-plane is
(
0
,
4
,
−
1
)
.
The projection of
(
−
2
,
4
,
−
1
)
onto the
xz
-plane is
(
−
2
,
0
,
−
1
)
.
Correct Answers:
•
-2
•
4
•
0
•
0
•
4
•
-1
•
-2
•
0
•
-1
Problem 2.
(1 point)
Determine
whether
the
three
points
P
= (
6
,
8
,
1
)
,
Q
=
(
7
,
10
,
4
)
,
R
= (
8
,
13
,
7
)
are colinear by computing the distances
between pairs of points.
Distance from
P
to
Q
:
Distance from
Q
to
R
:
Distance from
P
to
R
:
Are the three points colinear (y/n)?
Solution:
SOLUTION:
Distance from
P
to
Q
:
p
(
7
−
6
)
2
+(
10
−
8
)
2
+(
4
−
1
)
2
=
1
√
14
Distance from
Q
to
R
:
p
(
8
−
7
)
2
+(
13
−
10
)
2
+(
7
−
4
)
2
=
√
19
Distance from
P
to
R
:
p
(
8
−
6
)
2
+(
13
−
8
)
2
+(
7
−
1
)
2
=
√
65
In order for the points to lie on a straight line, the sum of the
two shortest distances must equal the longest distance.
Since
1
√
14
+
√
19
̸
=
√
65, the three points do not lie on a straigh line.
Correct Answers:
•
3.74165738677394
•
4.35889894354067
•
8.06225774829855
•
N
Problem 3.
(1 point)
What is the distance from the point
(
6
,
3
,
−
2
)
to the xz-plane?
Distance =
Solution:
SOLUTION:
The distance from the point to the
xz
-plane is the absolute value
of the y-coordinate of the point. Thus the distance is 3.
Correct Answers:
•
3
1
Problem 4.
(1 point)
What do the following equations represent in
R
3
?
Match the two sets of letters:
a. a vertical plane
b. a horizontal plane
c. a plane which is neither vertical nor horizontal
A.
−
5
x
+
8
y
=
1
B.
x
=
−
3
C.
y
=
−
5
D.
z
=
−
9
Correct Answers:
•
A
•
A
•
A
•
B
Problem 5.
(1 point)
Find the equation of the sphere centered at
(
−
1
,
8
,
−
7
)
with radius
6.
= 0.
Give an equation which describes the intersection of this sphere
with the plane
z
=
−
6.
= 0.
Solution:
SOLUTION:
An equation of the sphere with center
(
−
1
,
8
,
−
7
)
and radius 6 is
(
x
+
1
)
2
+(
y
−
8
)
2
+(
z
+
7
)
2
=
6
2
or
(
x
+
1
)
2
+(
y
−
8
)
2
+(
z
+
7
)
2
−
6
2
=
0 .
The intersection of this sphere with the plane
z
=
−
6 is the set of
points on the sphere whose z-coordinate is
z
=
−
6. Putting
z
=
−
6
into the equation yields
(
x
+
1
)
2
+(
y
−
8
)
2
+
1
−
6
2
=
0 or
(
x
+
1
)
2
+(
y
−
8
)
2
−
35
=
0
.
This is a circle in the plane
z
=
−
6 with center
(
−
1
,
8
,
−
6
)
and
radius
√
35.
Correct Answers:
•
(x - -1)**2 + (y - 8)**2 + (z - -7)**2 - 6**2
•
(x - -1)**2 + (y - 8)**2 + 1 - 6**2
Problem 6.
(1 point)
Find the equation of the sphere if one of its diameters has end-
points
(
2
,
−
1
,
10
)
and
(
4
,
3
,
16
)
.
= 0.
Solution:
SOLUTION:
The center of the sphere is the midpoint of the diameter:
(
2
+
4
2
,
−
1
+
3
2
,
10
+
16
2
)
= (
3
,
1
,
13
)
.
The radius is half the diameter, so
r
=
1
2
p
(
4
−
2
)
2
+(
3
+
1
)
2
+(
16
−
10
)
2
=
1
2
√
56.
Therefore an equation of the sphere is
(
x
−
3
)
2
+(
y
−
1
)
2
+(
z
−
13
)
2
−
56
4
=
0
Correct Answers:
•
(x - 3)**2 + (y - 1)**2 + (z - 13)**2 - 3.74165738677394**
Problem 7.
(1 point)
Find an equation of the sphere that passes through the origin and
whose center is
(
4
,
−
5
,
−
6
)
.
= 0
Note that you must put everything on the left hand side of the
equation and that we desire the coefficients of the quadratic terms
to be 1.
Solution:
SOLUTION:
The radius of the sphere is the distance from the center to the
origin:
r
=
√
4
2
−
5
2
−
6
2
=
√
77. Therefore the equation of the
sphere is
(
x
−
4
)
2
+(
y
+
5
)
2
+(
z
+
6
)
2
−
77
=
0
Correct Answers:
•
xˆ2 + yˆ2 + zˆ2 + (-8*x + 10*y + 12*z)
2
Problem 8.
(1 point)
Find an equation of the largest sphere with center
(
8
,
6
,
8
)
that is
contained completely in the first octant.
= 0
Note that you must move everything to the left hand side of the
equation that we desire the coefficients of the quadratic terms to
be 1.
Solution:
SOLUTION:
The largest sphere contained in the first octant must have a radius
equal to the minimum distance from the center
(
8
,
6
,
8
)
to any of
the three coordinate planes.
The shortest distance is 6, thus an
equation of the sphere is
(
x
−
8
)
2
+(
y
−
6
)
2
+(
z
−
8
)
2
−
6
2
=
0
Correct Answers:
•
xˆ2 + yˆ2 + zˆ2 - 2*(8*x + 6*y + 8*z) - 6ˆ2 + (8ˆ2 + 6ˆ2 +
8ˆ2)
Problem 9.
(1 point)
Find the center and radius of the sphere
x
2
−
20
x
+
y
2
+
4
y
+
z
2
+
10
z
=
−
128
Center: (
,
,
)
Radius:
Solution:
SOLUTION:
Completing the squares in the equation gives
(
x
2
−
20
x
+
100
)+(
y
2
+
4
y
+
4
)+(
z
2
+
10
z
+
25
) =
−
128
+
100
+
4
+
25
⇒
(
x
−
10
)
2
+(
y
+
2
)
2
+(
z
+
5
)
2
=
1,
which we recognize as an equation of a sphere with center
(
10
,
−
2
,
−
5
)
and radius 1.
Correct Answers:
•
10
•
-2
•
-5
•
1
Problem 10.
(1 point)
Write down an (in)equality which describes the solid ball of ra-
dius 5 centered at
(
−
10
,
8
,
2
)
.
It should have a form like
x
2
+
y
2
+
(
z
−
2
)
2
−
4
>
=
0, where you use one of the following symbols
≤
,<,
=
,
≥
,>
.
The first blank is for the algebraic expression; the drop-down
list gives the (in)equatilty.
?
0.
Solution:
SOLUTION:
The solid ball consists of all the points on or inside the sphere
with radius 5 and center at
(
−
10
,
8
,
2
)
. This set of points is de-
scribed by the inequality
(
x
+
10
)
2
+(
y
−
8
)
2
+(
z
−
2
)
2
≤
25, or,
equivalently,
(
x
+
10
)
2
+(
y
−
8
)
2
+(
z
−
2
)
2
−
25
≤
0.
Correct Answers:
•
(x - -10)**2 + (y - 8)**2 + (z - 2)**2 - 5**2
•
<=
3
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Problem 11.
(1 point)
You are given the following points:
A
= (
12
,
3
,
−
19
)
,
B
=
(
−
19
,
0
,
−
6
)
,
C
= (
−
10
,
−
20
,
17
)
.
Which point is closest to the yz-plane? [?/A/B/C]
What is the distance from the yz-plane to this point?
Which point is farthest from the xy-plane? [?/A/B/C]
What is the distance from the xy-plane to this point?
Which point lies on the xz-plane? [?/A/B/C]
Solution:
SOLUTION
The distance from a point to the
yz
-plane is the absolute value of
the
x
-coordinate.
The point
C
(
−
10
,
−
20
,
17
)
has the
x
coordinate with the smallest
absolute value, so
C
is the point closest to the
yz
- plane.
The distance from the
yz
-plane to
C
is given by the absolute value
of the
x
-coordinate, i.e.
|−
10
|
=
10.
The distance from a point to the
xy
-plane is the absolute value
of the
z
-coordinate.
The point
A
(
12
,
3
,
−
19
)
has the
z
coordinate with the largest ab-
solute value, so
A
is the point farthest from the
xy
- plane.
The distance from the
xy
-plane to
A
is given by the absolute value
of the
z
-coordinate, i.e.
|−
19
|
=
19.
A point lies on the
xz
-plane if its
y
-coordinate is zero.
Thus
B
(
−
19
,
0
,
−
6
)
lies on the
xz
-plane.
Correct Answers:
•
C
•
10
•
A
•
19
•
B
Problem 12.
(1 point)
Find the distance from
(
−
7
,
8
,
−
10
)
to each of the following:
1.
The
xy
-plane.
Answer:
2.
The
yz
-plane.
Answer:
3.
The
xz
-plane.
Answer:
4.
The
x
-axis.
Answer:
5.
The
y
-axis.
Answer:
6.
The
z
-axis.
Answer:
Solution:
SOLUTION
1. The distance from a point to the
xy
-plane is the absolute value
of the
z
-coordinate of the point. Thus, the distance is
|−
10
|
=
10.
2. The distance from a point to the
yz
-plane is the absolute value
of the
x
-coordinate of the point. Thus, the distance is
|−
7
|
=
7.
3. The distance from a point to the
xz
-plane is the absolute value
of the
y
-coordinate of the point. Thus, the distance is
|
8
|
=
8.
4.
The point on the
x
-axis closest to
(
−
7
,
8
,
−
10
)
is the point
(
−
7
,
0
,
0
)
, (Approach the
x
-axis perpendicularly.)
The distance from
(
−
7
,
8
,
−
10
)
to the
x
-axis is the distance be-
tween these two points:
p
(
−
7
+
7
)
2
+(
8
−
0
)
2
+(
−
10
−
0
)
2
=
p
(
8
)
2
+(
−
10
)
2
=
√
164
5.
The point on the
y
-axis closest to
(
−
7
,
8
,
−
10
)
is the point
(
0
,
8
,
0
)
, (Approach the
y
-axis perpendicularly.)
The distance from
(
−
7
,
8
,
−
10
)
to the
y
-axis is the distance be-
tween these two points:
p
(
−
7
−
0
)
2
+(
8
−
8
)
2
+(
−
10
−
0
)
2
=
p
(
−
7
)
2
+(
−
10
)
2
=
√
149
6.
The point on the
z
-axis closest to
(
−
7
,
8
,
−
10
)
is the point
(
0
,
0
,
−
10
)
, (Approach the
z
-axis perpendicularly.)
The distance from
(
−
7
,
8
,
−
10
)
to the
z
-axis is the distance be-
tween these two points:
p
(
−
7
−
0
)
2
+(
8
−
0
)
2
+(
−
10
+
10
)
2
=
p
(
−
7
)
2
+(
8
)
2
=
√
113
4
Correct Answers:
•
|-10|
•
|-7|
•
|8|
•
sqrt(8ˆ2+(-10)ˆ2)
•
sqrt((-7)ˆ2+(-10)ˆ2)
•
sqrt((-7)ˆ2+8ˆ2)
Problem 13.
(1 point)
Match the equations of the plane with one of the graphs below.
A
B
C
D
E
F
1.
x
+
z
=
2
2.
x
+
y
=
−
2
3.
x
+
y
=
2
4.
y
−
x
=
2
Note:
You can click on the graphs to enlarge the images.
Solution:
SOLUTION
1. The plane
x
+
z
=
2 is a plane parallel to the y-axis, that in-
tersects the
x
-axis at the point
(
2
,
0
,
0
)
and the
z
-axis at the point
(
0
,
0
,
2
)
. Thus the equation matches the graph D.
2. The plane
x
+
y
=
−
2 is a vertical plane that intersects the
xy
-
plane in the line
y
=
−
2
−
x
. Thus the equation matches the graph
C.
3. The plane
x
+
y
=
2 is a vertical plane that intersects the
xy
-
plane in the line
y
=
2
−
x
. Thus the equation matches the graph
A.
4. The plane
y
−
x
=
2 is a vertical plane that intersects the
xy
-
plane in the line
y
=
2
+
x
. Thus the equation matches the graph
B.
Correct Answers:
•
D
•
C
•
A
•
B
5
Problem 14.
(1 point)
Match the equations of the spheres with one of the graphs below.
A
B
C
D
E
F
1.
x
2
+
y
2
+
z
2
=
4
2.
x
2
+
y
2
+(
z
+
1
)
2
=
9
4
3.
(
x
−
1
)
2
+(
y
−
1
)
2
+
z
2
=
1
4.
x
2
−
2
x
+
y
2
+
2
y
+
z
2
−
2
z
=
−
2
Note:
You can click on the graphs to enlarge the images.
Solution:
SOLUTION
1. The sphere is centered at the origin and it has radius 2. Thus it
matches A.
2. The sphere is centered at
(
0
,
0
,
−
1
)
and has radius
3
2
. Thus it
matches C.
3.
The sphere is centered at
(
1
,
1
,
0
)
and has radius 1.
Thus it
matches B.
4. Completing the squares, yields
(
x
−
1
)
2
+(
y
+
1
)
2
+(
z
−
1
)
2
=
1. Thus the sphere is centered at
(
1
,
−
1
,
1
)
and has radius 1. Its
graph matches E.
Correct Answers:
•
A
•
C
•
B
•
E
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6
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