SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 1
Problem 1.
(20 points)
(a)Find an equation of the largest sphere with center (5
,
4
,
9) that is contained in the first
octant.
(b)Find an equation for the surface consisting of all points
P
for which the distance from
P
to the
x
axis is twice the distance from
P
to the
yz
plane. Identify the surface.
Solution:
(a) Imagine that a sphere is growing up from the neighborhood of that point.
Which
coordinate plane is the first one that is contacted by this growing sphere? It is the nearest
coordinate plane, in this case, the
xz
plane. At this time the radius of the sphere is 4. So,
an equation of this largest admissible sphere is
(
x
−
5)
2
+ (
y
−
4)
2
+ (
z
−
9)
2
= 16
.
(b) Let the coordinates of
P
is (
x, y, z
), then the distance from
P
to
x
axis is
radicalbig
y
2
+
z
2
.
The distance from
P
to
yz
plane is

x

. So the equation for the set of all points satisfying
the condition is
radicalbig
y
2
+
z
2
= 2

x

.
Square both sides and write in the standard form,
y
2
+
z
2
= 4
x
2
.
We identify this surface as a
cone
.
Problem 2.
(20 points) A clothesline is ted between two poles, 8 m apart.
The line is
quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the
middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the
clothesline.
Solution:
1
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SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 1
0.8 x 9.8 N
8 cm
4 m
4m
x
y
θ
θ
θ
θ
T
T
The angle
θ
is determined by
tan
θ
=
8 cm
4 m
=
0
.
08
4
= 0
.
02
.
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 Spring '08
 Perutz
 Calculus, CALCULUS III SAMPLE, b1 c1 d2

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