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AP Calculus
Related Rates
. ..
Related Rates
1.) The radius "r" of a sphere is increasing at a rate of 2
inches per minute. Find the rate of change of the volume
when r = 6 inches and r = 24,iflches.
dr
_I")~
'dt-
- c:
w.
"",n
~
dv
I \
"'Jb ~ ~
(-=
~
~f
':. Y
7..
-nr'C
dv
d",("
clt
~
~1ll~~) (7,.')
lP
(it
= c1~
i()'3
::&
21¢€>1T rv;=;-fl
L.Mr(;24~)l~) -::.~~=-%og1r~
r~2L..\
~__.
2.)
Ripples in a Pond: A pebble is dropped into a calm pond,
causing ripples in the form of concentric circles. The
radius "r" of the outer ripples is increasing at a constant
rate of 1 foot per second. When the radius is 4 feet, at
what rate is the total area "A" of the disturbed water
changing? _
A .¢If'{ '2
--
,
3.)
__ (t11
"r:.\ -- \~~
d
~
s e c,
-
r~ 4-~~+
~.
_ .~ ('-\)t,) : &r ~
d.~ -
dPt.
d\:;
~
om-'f"' 9!:
d..-t..
a..
O£'l
..'
SaL.
All edges. of a cube are expanding at a rate of 3
. centimeters per second. How fast is the volume changing
when each is 10 centimeters?
d'lG
dl
--
-
3c:xn
~e..c..
"
"
.
4.) Find the rate of change of the volume of a cone if the
rate of change of the radius is 2 inches per minute and
h = Jr when r = 6 inches.
v=
dv =.?
dt
.
c:ir
<it
= cD)JI1:":
m)r,
~"fT"('
V::
~1T'
V -=
1)'<
z. h
yo" ('0(')
'!t
d.v
2. ci~
cL-t, ~ 3~(' "~
h=or
~,
dv
(,
dt
: ~~('=bI..t:J)~
')
_
- ~\lon-
,/~
(5.)'
A conical tank (with vertex down) is 12 feet across and
~
10 feet deep. If water is flowing into the tank at a rate of ,
5 cubic, feet per minute, find the rate of change of the
depth of the water when the water is 6 feet deep?
= 5-!!.:
d\J.
dt
~\'"
dh
dX ~ '?
\j:
:~\ \J,
-~'
,_.\
J:._Je
h ~'"
10
': \0
r
-_.r i-±-h
5
~"t\("
•••
~
a..
'Q
'Io ••• ~.
zh.,
t.,,-;.
~h h
e;;
2.
,In'
;;;'1')
•
-
;
A rocket, rising vertically ~is tracked by a radar station
that is on the ground 3000 feet from the launching pad.
At what rate is the angle of elevation changing when the
rocket is 4000 feet up and rising vertically at 5000 feet
per second? dh
~+J
dt :::sCOO'
I!::"e.~
h~40()D
~
-rf\ to...) e :::.'300-t:>
<3e,c 2.. G c.:i§ -:. ...1... d h '
'3 000
d\:.
~c.
2.
aooo
d&
S qg
n :::
~w.·
~q?1~
.4:~ ~}7.)
ci.-1:. '
.
.~..~...
~.Ql.dk~c.
Air is blown into a ~herical balloon. How fast is the
radius changing whenthe air is blown into the balloon at
the rate of 10 centimeters per second at the instant the
. radius is 5 centimeters?'
,
-
dr
_?
d-t - .
.2:';L _
I D~1I
dt - - se "'"'
.
r::
is
CX'Y'"
\/ ~~ r~
ct.\J
'a- ~
---,
dJc - ":
- y.~- -. ~--- .
}b ;: 4 -rr ('2.-s) ~~
~~_
cLr
IO'Tl ?f.'-
-cc-~
•
7sV
A weather balloon is rising up from the ground at the
L / rate of 140 feet per minute. This. balloon is being tracked
by radar located on the ground 500 feet from the point of
lift-off. Find the rate of change of the angle of elevation
when the balloon is 500 feet above the ground.
tin
~
""
~
=\4(:) ~
de-
~
-'k
av1-
~.~
:=..
So (:)
n
:
f:oo
@
-=- '1fJUt
o« L\S 0
h
9 :::~
~Ct,n
de'
\ ( "
G-t -:.-L
~on <1h
-d.-!b
Se.c..2.(:?~) (;;u-)
=- Soo \~(),
An airplane is flying on a flight path with an altitude of 6 ~ ~ 1.~
miles that will take it directly over a radar tracking
cde 51) tI
station. If Us" is decreasing. at a rate of 400 miles per
hour when s' = 10 miles, what is the speed of the plane?
~"A
.
9.)
:=:
<Xl-
-
-
==::>
~,~
\::7
~
d~ _-400ml~
dt -
lLJ
ho\..U""
t::: \()('()\
\eA--
0,
•
,,'
10.) A television camera at ground level-is filming the lift-off
of a space shuttle that is rising vertically according to the
position equation, s = 50t2, where "s" is measured in feet
and "t" is measured in sec.onds.The camera is 2000 feet
from the .launching pad. Find the rate of change in the
angle of elevation of the camera at 10 seconds after lift off.
t .o
1':-
~'Q.t ~
+-a.n.&
c
a~eh
l1 o
~g~
_ ~- )
~c..
d-b - [}..o:t.
S
c: b
~(,
, \ n) ~
1\,11',
d-t.
"/
,
c'
';
-- 8
\ ',:-'
,
d%t ~ o. c-~q
rd~ec...
11.) A trough is-12 feet long and 3 feet across. It ends are
isosceles triangles with altitudes' of 3 feet.
a.)
If water is being pumped intothetrough at 1 cubic
.feet per minute" how--raSt-isthe water level 'rising
when "h" ~s1 foot deep?
'J ~ {( cU6)'rv
'I':: \ (cl b) \~
b).
j;!:- \ wd
1%
'*3 ::\~(\). ~
clb
, ~~/~,
L:\f~li·
t e water IS nsmg at a rate 0 ~r
\J ~ (p(a b,)
If"-h~'to dl.. ..
_ dt!
d~
trotA -
mlnute
when h = 2, determine the rate at which wa~er is
·
d·
h.
gh
belng
pumpe Into t ,e trou .
a" ~1'9,.d-~~
,'.' ~
~~Jt;
~-l;,
<
ct\:- d2 tOl),cti\,.
,'/,.' J
~
dv
"db
3, ()~3J
= 1,,\"1' tir« {\
\"2..,,,
-
=
3
gitK.h
'H
,~ ~
. .
if
IG) A ladder 25 feet long is leaning
l7
a.)
against the wall of a
house. The base of the ladder is pulled away from the
wall at a rate o(~t-per
second.
How fast is the top of the ladder moving down the wall
when its base is 7 feet, 15 feet and 24 feet from the wall?
~
d(,s)L~)~
~?
~t
.
(Po
dlW)( ~)-:.
+ ~'O ~/.L:t = 0
~/di
X '7. + 'j'Z.~2S 2
I")
c!~
cl-lC.
()(~~
t
dJ ~
~
0
':
d (,)(~) ~ ;zOix. <ll.t- )
"~l6)4)l~)
~
0
Ib.)J
V
1'2.
..~
..:2C1x.. 7d~
':.. 0
uc :
.'
t.i,c.
':..- 3/2 -Pt-Is<!<~I
.-PH
(1' Is~(..,
_ If 8)
Consider the triangle formed by the side of the house, the
ladder, and the ground. Find the rate at which the area of
the triangle is changing when the base ,of the ladder is 7
....
1
fee~f!:onUhe wall.
dA.. -.fU/,!"}' .~Ii. ~ ...
,.x)_"/~fI~rJ;~ ~~
~
- d '61'5:.-~J
n~'~~~
)'J;.
-=- ~ x :
~
7-.
:s: .
.
"'--AI
d
'-- /
~I,
".,; ::' ~;2S
~ 't-
I
Q..t
l-
d 1-
~.,.
f"(:,,:..,..,~_ft-F ~
-"i.
.
r' " 'f. /) '-\~ ..~ J
-'i- ({-lS~'- Y~) '- ~;~L~~ ~) 7~~ ..,j ~ .•<i~:~:~
y~", ~ '>-~
c.)
+
..
~ _ -J. ~J
dt -
0
~,=~~~~x"')\~~
u(
I
- - ~[4~-~1~1
.'
q(o.f.tJerc,c,
Find the rate at which the angle between die ladder and
the wall of the house is changing when the base of the
ladder is 7 feet from the wall.
5lrv f} -=
:L1---)
eke-
)
(jQ 5 ~ ~
/\ ~
e-. ~
~J
\j
dt
~
_'
-
~
;-?' ;~~:
-
.-L ('2.} ._.
.',,'Z-S"
'.
de
-at
~~,cadi-~ft;;,~~···.
"
~.;
.
13.) The radius "r" of a sphere is increasing at a constant rate
of 0.04 centimeters per secon-a~---..
a.)
At the time when the radius of the sphere is 10
centimeters, what is the rate of increase of its
volume?
V==- ~
11' $,
.
~
::: 111i"Y z ~
dv
"M.
=
4-cr ( \ 0 0) o· 0,"+
~ = \ lo1r
CJY'/
see:
At the timewhen the volume of the sphere is 361r
cubic centimeters, what is the rate of increase of the
area of a cross section through the center of the
sphere?
.
u
&..
b.)
3loTT :::: '3 '1fY
,\
3
3<..0 ~3 '<"'
l' .
'2..
..~..C\ =1TY
<".'
~.
d A- ~:;hJY<l!:
db -
~A
~
::; <.a1l
~,..
c.)
(()"7I::::
d~
v~
cA"
3~~
(o.ou.)
0 q,\{'fj" -'>
I
cvtlse.c.
~?J
At the time when the volume and the radius of the
sphere are increasing at the same numerical rate,
what is the radius?
,dv
c~
a;
c:U:
,·L
,\
r
r. ·a-t·, '. .:.~'.
G;~:'
Oll
41rf 2.~~)' =, 0,:.
41i('~o,d4)--~ 0
~
"1
1~~
\.,
,--::;.,
\ .)
~, '" 3 5\1. A
,~~~-~
':..'r= B~ IV • ,
'1-
M
(...A'\
. ..
"
•
.
,
14.) Sand is falling off a conveyor onto a conical pile at the
rate of 15 cubic feet per minute. The diameter of the base
of the cone -isapproximately twice the altitude. At what
rate--------is the height of the pile changing when it is 10 feet
high?
-ft-.3,.
4Y,..
dt
- \S-
~
\
\_~
V~ ~
1\,n
,,1:- ~
ll'h 3
\...:..
n
d"
'1rh' sib
~
::
clt
\S
~
-:::.."U'"'
\
cl-'"
00 ~
~-TJ.
~
I~
~o~
<!b..
dl~
»: .----',
(' 15.) }\ balloon rises at the rate of8 feet per se~~nd from point
J on the ground 60 feet from an observer. Find the rate of
-change of the angle of elevation
when the balloon isr-------25
feet above the ground <, __
:~~&h~,d.e-
-::::?
c:L~
"'- ~
"L
S-'
-,
..,.:;..'