Part II: The Answers
288
Answers
401–500
442.
4
25
ft/min
The water in the trough has a volume of V
b h
= 1
2
20
( )( ). As the water level rises, the
base and height of the triangle shapes increase. By similar triangles, you have 5
2
= b
h
so
that b
h
= 5
2
. Therefore, the volume is
V
b h
V
h h
V
h
=
=
=
1
2
20
10 5
2
25
2
( )( )
( )
Taking the derivative with respect to time gives you
dV
dt
h dh
dt
= 50
Use h = 1 foot and dV
dt
= 8 cubic feet per minute to find how quickly the water level is
rising:
8 50 1
8
50
4
25
=
=
=
( ) dh
dt
dh
dt
ft/min
When the water is 1 foot deep, the water level is rising at a rate of 4
25
feet per minute.
443.
698.86 km/h
Let x be the distance that the jet travels, and let y be the distance between the jet and
the radar station.
Use the law of cosines to relate x and y:
y
x
x
y
x
x
y
x
x
2
2
2
2
2
2
2
2 2 2
1 35
4 4
2
2
2 2
4
=
+ −
°
=
+ −
−
=
+
+
( ) cos(
)
288
Answers
401–500
442.
4
25
ft/min
The water in the trough has a volume of V
b h
= 1
2
20
( )( ). As the water level rises, the
base and height of the triangle shapes increase. By similar triangles, you have 5
2
= b
h
so
that b
h
= 5
2
. Therefore, the volume is
V
b h
V
h h
V
h
=
=
=
1
2
20
10 5
2
25
2
( )( )
( )
Taking the derivative with respect to time gives you
dV
dt
h dh
dt
= 50
Use h = 1 foot and dV
dt
= 8 cubic feet per minute to find how quickly the water level is
rising:
8 50 1
8
50
4
25
=
=
=
( ) dh
dt
dh
dt
ft/min
When the water is 1 foot deep, the water level is rising at a rate of 4
25
feet per minute.
443.
698.86 km/h
Let x be the distance that the jet travels, and let y be the distance between the jet and
the radar station.
Use the law of cosines to relate x and y:
y
x
x
y
x
x
y
x
x
2
2
2
2
2
2
2
2 2 2
1 35
4 4
2
2
2 2
4
=
+ −
°
=
+ −
−
=
+
+
( ) cos(
)