Homework 3 Solutions
12.1.2 The distance of a point
P
= (
x, y, z
)
from the
yz
plane is

x

, from the
xz
plane
is

y

, and from the
xy
plane is

z

.
So
A
is closest to the
yz
plane, since it
has the smallest
x
coordinate in absolute value.
B
lies on the
xz
plane, since
its
y
coordinate is 0.
C
is farthest from the
xy
plane, since it has the largest
z
coordinate in absolute value.
12.1.6 The distance formula
d
=
p
(
x
2

x
1
)
2
+ (
y
2

y
1
)
2
+ (
z
2

z
1
)
2
, gives us the
distance between any pair of points
(
x
1
, y
1
, z
1
)
and
(
x
2
, y
2
, z
2
)
. Thus, we find
Distance from
P
1
to
P
2
=
2
√
2
Distance from
P
2
to
P
3
=
√
6
Distance from
P
1
to
P
3
=
√
10
So
P
2
and
P
3
are closest to each other.
12.1.22
(a) The total cost
C
in dollars of renting a car is 40 times the number of days
d
plus 0.15 times the number of miles driven
m
. So
C
=
f
(
d, m
) = 40
d
+ 0
.
15
m.
(b) We have
f
(5
,
300) = 40(5) + 0
.
15(500) = $245
.
Renting a car for 5 days and driving it 300 miles costs $245.
12.1.31 The edges of the cube have length 4. The center of the cube is the point
(
x, y, z
)
where the coordinates are found by taking the midpoint of the edges of the cube:
x
=
2 + 6
2
= 4
,
y
=
5 + 9
2
= 7
,
z
=

1 + 3
2
= 1
.
Thus, the center of the sphere is the center of the cube which is the point
(4
,
7
,
1)
and the radius is
r
= 2
. Thus an equation of the sphere is
(
x

4)
2
+ (
y

7)
2
+ (
z

1)
2
= 4
.
12.2.2
(a) The value of
z
only depends of the distance from the point
(
x, y
)
to the
origin. Therefore, the graph has a circular symmetry around the
z
axis.
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 Fall '07
 Hohnhold
 Calculus, $245, 5 days, 0.15M, 0 4 m

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