1
ma2176 a3 sol
Applications of Derivatives
1.
An airplane, flying horizontally at an altitude of 1 km, passes directly over an observer. If the constant
speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 seconds
later?
Solution
:
h
km
1km
s
km
The relation between
h
and
s
is
2
2
1
h
s
+
=
.
Differentiate both sides of
2
2
1
h
s
+
=
with respect to
t
, we have
dt
dh
s
h
dt
ds
dt
dh
h
dt
ds
s
=
⇒
=
2
2
.
Since
km
2
120
1
240
km/hr,
240
=
=
=
h
dt
dh
and
5
5
2
1
2
2
=
⇒
=
+
=
s
s
.
Thus
km/hr
5
480
=
dt
ds
.
2.
Water is pumped at a uniform rate of 2 liters per minute into a tank shaped like a frustum of a right
circular cone. The tank has altitude 80 centimeters, and its lower and upper radii are 20 and 40
centimeters respectively. How fast is the water level rising when the depth of the water is 30
centimeters? (Hint: The volume
V
of a frustum with altitude
l
and lower and upper radii
a
and
b
is
()
22
1
3
Vl
a
a
b
b
π
=+
+
)
Solution
:
frustum
water
h
20
2
x
(removed part)
Suppose the depth of water is
h
and the radius of
the surface of water is
d.
Using properties of similar triangles, we have
x
x
h
d
+
=
20
.
In addition, we know that the tank has altitude 80 centimeters and lower and upper radii of 20 and 40
centimeters respectively. Using properties of similar triangles again we have
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80
1
80
2
80
20
40
=
⇒
+
=
⇒
+
=
x
x
x
x
Thus, we have
20
4
80
80
20
+
=
⇒
+
=
h
d
h
d
.
The relation between the volume of water
V
and the depth of water
h
is :
()
22
1
3
Vh
d
d
b
b
π
=+
+
, where
d
is the radius of
the surface of water and
b
is lower radius of the frustum.
.
Since
20
,
20
4
=
+
=
b
h
d
, we have
2
1
20
20
20
400
34
4
hh
+
+
+
.
Now differentiating
V
with respect to
t
gives
2
11
1
1
20
20
20
400
2
20
20
.
4
3
4
4
4
dV
dh
h
h
h
dh
dh
h
dt
dt
dt
dt
ππ
+
+
+
+
+
+
Put
2000 and
30
dV
h
dt
==
into the above identity, we have
2
13
0
3
0
1
3
0
1
1
2000
20
20
20
400
30 2
20
20
4
3
4
4
4
1
2000
756.25 550 400
10
13.75
5
568.75
187.5
3
2000
cm
min
756.25
dh
dh
dh
dt
dt
dt
dh
dh
dh
dh
dh
dt
dt
dt
dt
dt
dh
dt
+
+
+
+
+
+
⇒=
+
+
+
+
=
+
3.
A particle
P
is moving along the graph of
2
,
4
2
≥
−
=
x
x
y
, so that the
x
coordinate of
P
is
increasing at the rate of 5 units per second. How fast is the
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 Spring '08
 LUI
 Trigraph, Orders of magnitude, Natural logarithm, DT DT DT

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