Section 4.6: Related Rates
l Dc~':1 i)
Homework: 3-36 mult. of 3, 34
**Related Rate Equations
How fast is a balloon rising at a given instant? How fast does the water level drop when a tank is
drained at a certain rate?
Differentiate to get an equation that relates the rate we seek to the rates we know.
(Implicit Differentiation x 2) ©
Related Rate Problem Strategy
1. DmU' a p i IlIrl" alld lIame Ihe var iables and OI/Sl allls. Use I for
time. As ume aU mabie art differentiable function of I.
2. Wrile down the I/ll merico f ill/ormation (in terms of th ,ymbols
you have cho. en).
3. Wriu dOH'1l whor we ali a ked 10 filld (u. ually a rate. ex pres. ed as
a deri v:lli\'e).
4. Write n equalion Ilwl rel(l/t:l Ihe variuble . ~ u may have to com­
binl: (Wo or more quations l gel a si ngl equation thai relates (he
varia ble whose rale y u want to the vuri ab les who. e rate.! Y u know.
S. Differentiate lI'i llr
(erm ' of lht!
rt'lpeci Iv I. Then expre ~
the rate you want in
rate and variables wh e \alue~ ) ou know,
6. E'·olliate. U ' known value
(0 find the unknown rdte.
Example 1:
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?
~=rrc\
d:.
(' I ~
\) -:. \1
\0 \<\
dL " chCll'\gelY\
Ddieren1\a\ "
cle. o lin.
d)'. : C\\ Qn 9e.
ch. ':. <,
IV)
Q\ ame1e.y­
dG :: 11 c\x
Z-11 dx
dx= 2
if
d\Qme~er
b~ 21lf
chanses
in
\\now'
A~ n rl 211 y­
} \'
Differen1lQ\ ~ d~ ~
2TTr dr
7.
dr:i..d)c
~
2
1(~)
df\ ~ 2TT(5)(~)
dA= I = i
-IT
Qfeo
changes
bj
\0 ir/
Example 2: If a and b are the lengths of two sides of a triangle, and (J the measure of the included angle, the area A of the triangle is A =
G)~ .
How is dA/dt related to da/dt, db/dt, and d (J/dt? in p\e chc\\\\
r ~~
(\1 \e
i
f <)' \t
t
,~n
\
d~' li)bs,n0~ \ (i)c\s,!\e~~
t
(~)abcose~~
Examples 3:
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?
what
Know :
i Q dd er : 210 \
dx
cit c.ha",ge
X~ \()\
Il'\
hofllontal
directIon
y
dx
dL ~
4- ·-\t\seL
~ : chQnqe1n Vtr1ICC\\
dt
Glfec1\OY)
d~: ?
cit
\QZ -t ~l ~ 2~2
'3-=24\ ~:
24
P~1naCjorea\l 1hm", 2x dx + 2t ~
cit.
clt
-= 0
2). ~ = - 2~ d~ ------...~ .~ d). ~ _~ dy
dt
dt
<it
cit
10
l'1-)
0
­
12Y
1~~
d~ -5
-dt - -3 ft ·/sec .
Example 4:
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 radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep?
WhQl
KnoW ',
dV ':; 2 f"t 3 j
cit
5
h ~ 10
r~
V~
Qn
~11 r 2
:0
dt
N'lm
v~ ~ 11,111
V=
h
_I
i2
-">
~ TI (~)2 h
TIh 3
\\/=~lThL\
?
.
Example 5: A boat is being pulled into a dock by a rope with one end attached to the bow of the boat, &
the other passing 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 ftjsec, how fast is the boat approaching the dock
when 10 ft of rope are out?
Know , h "" 4-
1
\j"" \()
Ai
~ 2 Hisel.
dt
~~ ?
dt
'
d~
\0
:0
{2j -ft i .sec
dt
~
2, \82 ft./sec .