Related Rates Problem Set
(Revised March 8, 2011)
1.
#Þ
If D # œ B# € C# ß .BÎ.> œ #ß and .CÎ.> œ $, find .DÎ.> when
B œ & and C œ "#Þ Assume that D !Þ
A particle moves along the curve C œ È" € B $ Þ As it reaches
the point a#ß $bß the y-coordinate is increasing at a rate of
4 cm/sec. How fast is the x-coordinate of the point changing
at this instant?
3.
.T
Suppose T Z œ 5 where 5 is a constant and
œ # when T œ %
.t
.Z
and Z œ &Þ Find
Þ
.>
4.
Two cars start moving from the same point. One travels south at
60 m/hr and the other travels west at 25 m/hr. At what rate is
the distance between the cars increasing two hours later?
5.
A plane flying horizontally at an altitude of 1 mile and a speed
of 500 mph passes over a radar station. Find the rate at which
the distance from the plane to the station is increasing when it
is 2 miles away from the station.
6.
A conical water tower has a height of 12 ft and a radius of 3 ft.
Water is pumped into the tank at a rate of 4 ft$ /min. How fast is
the water level rising when the water level is 6 ft?
See the diagram.
3 ft
12 ft
7.
A ship is 40 miles west of a lighthouse. The ship is heading
north at a rate such that the angle ), shown in the diagram below,
is changing at a constant rate of !Þ( radians per hour. At what
rate is the distance B between the ship and the lighthouse
changing when ) œ !Þ% radians?
ship
x
40 miles
lighthouse
8.
At noon, ship A is 100 km west of ship B. Ship A is sailing south
at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is
the distance between the ships changing at 4:00 P.M.?
9.
An 20 ft long ladder is leaning against a wall. The bottom of the
ladder is sliding away from the wall at a rate of 2.5 ft/sec.
See the diagram.
20 ft
y
x
a) How fast is the top of the ladder sliding down the wall when
B œ "# ft. Note that this rate is |.CÎ.>l Þ
b) How fast is the angle ) changing when B œ 12 ft?
c) How fast is the area of the triangle changing when B œ 12 ft?
Solutions
1.
2.
If D # œ B# € C# then
.D
.B
.C
#D
œ #B
€ #C
Ê
.>
.>
.>
B.BÎ.> € C.CÎ.>
%'
a&ba#b € a"#ba$b
Dw œ
œ
œ
Þ
È&# € "##
D
"$
If C œ È" € B$ then
.C
.C .B
œ
†
Ê
.>
.B .>
.C
"
.B
•"Î#
ˆ$B# ‰
œ ˆ" € B$ ‰
Ê
.>
#
.>
%œ
%œ
3.
"
#a" €
#$ b"Î#
ˆ $ † ## ‰
.B
Ê
.>
"# .B
.B
Ê
œ # cm/sec .
' .>
.>
If T Z œ 5 then
.T Z
.5
œ
Ê
.>
.>
.T
.Z
.Z
Z €T
œ ! Ê #a &b € %
œ! Ê
.>
.>
.>
.Z
"!
&
œ •
œ • Þ
.>
%
#
4.
Look at the situation > hrs later, where > is an arbitrary time.
The 1st car has moved south (say a distance y) and the 2nd car
has moved west (say a distance x). We have the triangle below.
x
y
s
B# € C# œ =# o differentiate both sides with respect to >
BBw € CCw
#BB € #CC œ #== Ê = œ
=
where Bw œ #& and Cw œ '! and =w is the unknown in the
problem. Note that 2 hours later, B œ 50 mß C œ 120 m,
and = œ È&!# € "#!# œ "$! m. Therefore
a&!ba#&b € a"#!ba'!b
=w œ
œ '& m/hr.
"$!
w
w
w
w
5.
500 mph
s
1 mile
x
#
#
= œB €"
#
w
w
Ê #== œ #BB
#&!È$ œ %$$ mph.
a&!!bŠÈ% • "‹
BBw
Ê = œ
œ
=
#
w
6.
Let h be the height of the water lever and let r be the
radius of the cone of water.
3 ft
3
12 ft
r
12
h
"#
2
2
œ
Ê 2 œ %< Ê < œ
$
<
%
#
1 $
"
"
2
Z œ 1<# 2 œ 1Œ • 2 œ
2
$
$
%
%)
.Z
.Z .2
"
.2
.2
'%
'%
œ
†
œ
12#
œ% Ê
œ
œ
œ
.>
.2 .>
"'
.>
.>
12#
1a'b#
'%
"'
œ
ft/min .
$'1
*1
Because of similar triangles,
7Þ
We want
.B
.)
œ !Þ( radians/hr Þ
when
.>
.>
ship
x
40 miles
lighthouse
Look at the drawing and find an equation involving B and ).
%!
Ê B-9= ) œ %! Ê
B
Ê B œ %!=/- ) Þ
One equation is -9= ) œ
Bœ
%!
-9= )
Now differentiate both sides wrt >Þ
.
.
B œ Ð%! =/- )Ñ Ê Bw œ %! =/- ) >+8 ) † )w Ê
.>
.>
.B
œ %! =/- a!Þ%b>+8a!Þ%b † a!Þ(b œ "#Þ)& mph Þ
.>
8.
Start
t hours later
100 km
25 km/h
100 km
100 km
B
B
A
4 hrs later
y
s
35 km/h
240 km
A
=# œ C# € "!!#
w
#== œ #CC
w
CCw
Ê = œ
Þ
=
w
4 hrs later C œ %a#&b € %a$&b œ "!! € "%! œ #%! 57
Cw œ #& € $& œ '! 57Î2<
a#%!ba'!b
=w œ
œ &&Þ$)& 57Î2<
È#%!# € "!!#
9.
Triangle at arbitrary time
20
y
x
Triangle when x= 12
y
20
12 ft
a) B# € C# œ %!! Ê #BBw € #CCw œ ! Ê
• BBw
Cw œ
Þ
C
When B œ "#, C# € "%% œ %!! Ê C œ È%!! • "%% œ "' Þ
• a"#ba#Þ&b
Cw œ
œ • "Þ)(& ft/sec
"'
Therefore the top of the ladder is sliding down at a speed of
1.875 ft/sec.
b)
20
y
12 ft
C
.C
.)
Ê C œ #! =38 ) Ê
œ #! -9= )
#!
.>
.>
.)
Cw
• "Þ)(&
Ê
œ
œ
œ • !."&' radians/sec
.>
#! -9= )
#!a"#Î#!b
=38 ) œ
c)
Eœ
"
.E
" .B
" .C
BC Ê
œ
C€ B
œ
#
.>
# .>
# .>
"
"
a#Þ&ba"'b € a"#ba • "Þ)(&b œ )Þ(& ft# /sec .
#
#