Section 4.6: Related Rates In related rate story problems, the idea is to find a rate of change (with respect to time) of one quantity by using the rate of change (with respect to time) of a related quantity.
~__________________~~~~~~~~X2)
Procedure for Related Rate Problems
~
1.
Drawa figure (if necessary) and choose variables for all unknowns.
2.
Write what is given and what is to be found using your variables and ~ symbols.
3.
Write an equation relating the variables.
dl
(a) If a quantity is changing it must be represented with a variable letter.
(b) If a quantity is constant it must be represented with a number value.
(c)
variables.
Look for secondary relationships between quantities to reduce the number of
4. Implicitly differentiate both sides with respect to t.
5.
Substitute number values and solve .
Geometry Formulas
Right Triangle
a" + b2 = c 2 Circle
=nrc
C = 2nr
A
A =~bh
2
Sphere
4
Cube
,
V ="., nr
V =e
Cone
A =6e
A = 4r.r"
Rectangle
J
A =fw
V =~m·2h
P =21 + 211'
3
2
Examples:
1.
In this problem, the first three steps of the procedure are done. We need to complete
Given: dx
= -6 Y = 4 xy = 12
dl
'
'~
Find:
dy
"
f
dt
\J
4.
1 -= I
_\.
.
only the last two steps.
~ X~ = 12~
X= 3
ddf eren tlQte .
dx -0
ydt ­
3(~)i
(1)
(-t,)00
3 d:j/dt - 21-;. 0
d~/dt ~ 8 t ihe rate of change of
WIth
to t ({Ime)
2.
A windlass is used to tow a boat to a dock. The rope is attached to the boat at a point 15 feet below the level of
the windlass. If the windlass pulls in rope at the rate of 30 feet per minute, at what rate is the boat approaching the
dock when there is 25 feet of rope out? Write a sentence explaining the meaning of the answer.
~\JhQt
'1"
windlass
Know '
IS
d't. ,,- 3(\
d1.
\J
H (constant)
fir m.
If)
'1 ":. 25
)( 2.
~ :?
dt
''\ t
.
x'2 -t
i 51. t X "2 - 251.
lx d~
)\:20
dt
y"2 -= II
151
::
to=:
'2 (20) ~
=
1/'
21.- di­
dt
2(25) (-30)
oxdt : : -37.5
3.
Gravel is falling on a conical pile at the rate of 10
f-\:I
I~i:
.
Tn \t"\.
.
At all times, the radius of the cone is twice the height of
the cone. Find the rate of change of the height of the pile when the radius of the pile is 6 It. Write a sentence
explaining the meaning of the answer.
What
Know: dV '" 10 fF~/m'n
dt
r =- 7h (constOrri)
dh
--:::
7
dt
r -: : fo H.
Y= iTir2h
v~ ~\fr2h
4>
need on\'j 1 \iGnQble
V :: ~ if (2h )'1 h ~
3
V= 113 11 h
dV
1
c\h
cit - 1nh dt
\0:: 1n (3) 1 dn
dt
dh _ 5 H\ .
dt - ~ min
4.
The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much
did the tree' s diameter increase? The tree's cross section area?
What I know'
d-= lo~r":5
~: 1
dt
dd
:
(-=
~
C:= 7rrr
QI
dt
~~ ~ ~(5) l ~) 2 -IT ~
dt
d~
- -; \0 In!\.lr.
dd _ z .
dt ~ IT \°1 ~r.
5.
A '=- rrr 2
dA
dr
dt ' -: 7rrr dt
dd
:: IT
dt
?
Tid
at
J
A 26-ft ladder rests on horizontal ground & leans against a vertical wall. The foot of the ladder is pulled away
from the wall at 4 ft/sec. How fast is the top sliding down the wall when the foot is 10 ft . from the wall?
for dlreci\on(. .
-+ / -
it S~i ~ P
What
KnOw: d}. -= t 111
dt
r= 2b
"lI"
se.c .
l\\UlIg e Ih
OIl'eciIQ(I
(ccmsiuni )
'f-: \0
'f.,L·-t ) 1
= 1.. 2
AZ-t'Yl-:;2~1
Zx dx
d-t.
t
1 dy ': 0
~ dt
1 (to ) ( <f) t 2(z~) ~ " 0
dy
--5
dt:: 3
ft
Ise.c .
Water runs into a conical tank at the rate of 2 ft3jmin. The tank stands point down, has a height of 10 ft., & a base
6.
radius of 5 ft. How fast is the water level rising when the water is 6 ft . deep?
-r need
~
vanable
-
10
I. ~ ~
\
n
h-=-5
r~
5
~=7
<it
5h:: lOr
-4
10
r~
-4
h
2
V= ~ 1T (~)2 h~v: ,~ -rrh 3
V
d .~
cit
2. 1i h2 dh
dt
12
2 =;'11 (Iof dh
dt
4
dh = l:.- 11 (min
dt 9-rr
A boat is being pulled into a dock by a rope with one end attached to the bow of the boat, & the other passing
7.
thru a ring attached to the dock at a point 4 ft higher than the bow of the boat. If the rope is pulled in at the rate of 2
ft/sec, how fast is the boat approaching the dock when 10 ft of rope are out?
1
ftl s
h
WIJat
1=~ Iconsta ni ) )
dOcK
~~4
X,2+4 1 -_
2x d~_
:
)\=9bI5
dt
cit
?
ct~
dt
«t loll) ) ~ = 1 (10) ( 2 ) 2
dt
-;
1.
dx _'(
dt - . T
21
(Cons tant) 1": \0 d=tX2tyZ~t2
Know -·
2.0B
{f fsec .
Xli
'1..
2
t
yl -t 2
11 -= \01
8. A hot air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the
moment the range finder's elevation angle is ~, the angle is increasing at .14 radians/min. How fast is the balloon rising
4
at that moment?
~
tone -=.1
y ~ 500tGnG
500
dy 500 see 2ede
dt
dt
0
~O()
ft .
d~
'A ~
dt:c 500secz(iT/1) (.11)
know
WhQt
~oo (constQn t)
e -TT/i
~~ = 14
d~ ~
sec :: ~
o
~ 500
(2) ( . Ii)
~ 14-0 H{m/n .
7
cit
>
9. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in 3/min. How fast is the level in the
pot rising when the coffee in the cone is 5 inches deep?
whQt
Kno~
V=lir2h
dV ~ 10
dt
dh _<
dt - .
-
h~5
- -
~
r:: 3 (constant)
V=lTr 2 h
V~ IT (3)2 h
v:; qn h
dv -= grr dh
dt
cit
10: qn dh
dt
dh ~ \0 itll
dt qrr
mIn