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31. (a)
}
d
d
V
r
} 5 }
d
d
r
}
1
}
4
3
}
p
r
3
2
5
4
p
r
2
When
r
5
2,
}
d
d
V
r
} 5
4
p
(2)
2
5
16
p
cubic feet of
volume per foot of radius.
(b)
The increase in the volume is
}
4
3
}
p
(2.2)
3
2 }
4
3
}
p
(2)
3
<
11.092 cubic feet.
32.
For
t
.
0, the speed of the aircraft in meters per second
after
t
seconds is
}
2
9
0
}
t
. Multiplying by
}
360
1
0
h
sec
}?
}
1
1
00
k
0
m
m
}
,
we find that this is equivalent to 8
t
kilometers per hour.
Solving 8
t
5
200 gives
t
5
25 seconds. The aircraft takes
25 seconds to become airborne, and the distance it travels
during this time is
D
(25)
<
694.444 meters.
33.
Let
v
0
be the exit velocity of a particle of lava. Then
s
(
t
)
5
v
0
t
2
16
t
2
feet, so the velocity is
}
d
d
s
t
} 5
v
0
2
32
t
.
Solving
}
d
d
s
t
} 5
0 gives
t
5 }
3
v
2
0
}
. Then the maximum height, in
feet, is
s
1
}
3
v
2
0
}
2
5
v
0
1
}
3
v
2
0
}
2
2
16
1
}
3
v
2
0
}
2
2
5 }
v
6
0
4
2
}
. Solving
}
v
6
0
4
2
} 5
1900 gives
v
0
<
6
348.712. The exit velocity was
about 348.712 ft/sec. Multiplying by
}
360
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.
 Fall '08
 GERMAN

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