Related Rates

Implicitly differentiate !he following formulas with respect to time. State what each rate in the clifferei1tiai equation representS

I . A= 4nr

1

Surface Area of a Sohere 2.

V = 4nr3

�

Volume of a Sphere

� ., .

a= �c2 -b2 . where c is is constant

a.. :: ( c... 'a. -b '2.),�

4.

1· = r.r2h. where r is

constant

Volume of a Cvlinder 5.

cos8 = 1�-: 'fs><.

ts, 6.

V = l m-2h

'2-.JW

Volume of a Cone

1· !! - �� - t-rrr �

Qft -O,� - '?>'ti(' <!cd.t

i• � - 4:\ iT" (,. J.c...

d.t-d:t

£1_ :. "\"QT 2. d.r"ax a.�

������ de s��-. ... .

SL .., �( � �cJ..., c1.t

/\.o.JJ.v S 'ii, -' · • -"-�.�.., � cd-�J..."�' -d.� (. ,v -

ic,�

k ...,. � .! � �\l.ld..t: . c.\, • I��-��·

i cl.a. - .1.. ( c.1.-b'°)_,,�. -tb � �- � cl.t

<!1.. - -b

�t;--

J c."'-b,. �

6..o... o.,,J... � ...:.,d� d.b

�es � �t.A.- 0. � 'o o.,.a.

��· 1. � -d.-t - � e.b..ilr ci� c:l\J � J\A; t. 6 � ;�L.. \J� 11 ....

d.V � irf''l !bC,t" d..�

-S\V\� M. --cl�

\ -\,

,� dt . t:frh .s\b .. tuJ:� . '. cit- \'!. �-

�

� -'> N.l"A. � �l.�,L � �� ��d.�

t·� ���\� d.o.o�

d..\/ �nr,1-\ J..r -t �(" ... �- -::

� ci\ et

. ' u

� > fi I �� � .;t.a_ ("� °*" � <1.� � d

"�, J\..t.SUus, � J...� � �·

' I�

�

� Air is leaking out of an inflated balloon in the shape of a sphere at a rate o�

.,

3""

01t_c_\!bic centimeters per

minute. At the instant when the radius isetimeters, what is the@ of change otyie radn�f the

balloon? '(" �

I . Identify all of the variables involved in !he problem.

2. Identify which, if any, of the variables in !heproblem that remain cooslllDL

3. Identify !he nue(s) that are given and !he rate !hatyou wish to find.

4. Write an equation. often a geometric fonnula ortrigonometric equation, that relates all of !he variables in the problem for which a rate is given or for which a rate is to be determined. SUbstiture IIDl; 11iue ttia (l;jACSCIUS a v�ble that is coostan�th• jirobleiiil it is imporrant to keep in mind that you canhave only one more variable 1han you have razes- You may have to make a substitution that relates one variable in terms of another.

S. Implicitly differentiate both sides oflhe equationwith res-peCJ to lime.

6. Substitute all instantaneous rates and values of thevariable and solve for the remaining rate or variable.

�

,,-o..6..i\l s - i \J I- -

No o-!,� "ox\ .Jo\�s

�-�� -

�-;,o\f c.v,..1

1�

dr -: � J Cw"'- l l"M,t'\,, -

o.t

\)� �"Tf ,- l '!

* - 4r11r� if,-

-1301\ ::. '11T ( �)� rk d.t

-.lao'ti = ,4,r�

k. - - .l �o"A' ---3.SC\"\c1t

-

�-,f -

CM.\�

0

0

A stone is dropped into a calm body of water, causing ripples in the form of concentric circles. The radius

oftbe outer ripple is increasing at a rate of(I._ foot per seco� When the radius is@ at wba@e is the

total area)ftbe disturbed water changing? %

i

�1e

I. Identify all of the variables in,•olved in the problem.1 �-u.. � �us

2. Identify which, if any, of the variables in theNo \10..V;�l� c; �� problem that remain constanL

•

3. ldemify the rare(s) that are given and the rate !hat �r - .1.. �\�you wish to find. d.t -

6A -� I +t

l./ �

� -

4. Write an equation, often a geometric fonnula or /+-:.."Ttr 2.. trigonometric equation. that relates all of the variables

in the problem for "bich a rate is given or for which arare is to be determined. . Substitute 30) value that represents a ,·ariable that is constant throughout theproblem. Ir is imporrcmr 10 keep in mind thar you canhm•e 011/y one more 1•ariable 1h011 you hm·e rares. f oumay have 10 make a subsrinnion rhm re/mes one variable in rerms of anorher. 5. Implicitly differentiate both sides of the equation � - 2:�r�wirh respec1 10 rime. - -

ci.t

6. Substitute all instantaneous rares and values of the -;;(far 2:�( 4J( i)variable and solve for the remaining rate or variable. - � d,�

a.Pr :. l'8"tr -y\;�1�}-

�

* Water is leaking out of a cylindrical tank at a rate o 1 cubic feet per secon lf the radius of the tank is(l( � ar whm � the dsBtofthe water cffiin,ing al any instant during the leak?

�-t

I. Identify all oflbe variables invol\'ed in me problem.

2. Identify which, if any. of !he ,-ariables in !heproblem !hat remiun constant.

3. Identify the rate(s) rhm are ghen and the rate !batyou wish to find.

4. Write an equa1ion. often a geometric fonnula or rrigonometric equation. !hat relates all of me variablesin !he problem for which a rate is given or for which arare is to be detennined. Substitute anv value that re12resen1S a variable that is constant thW1U:IJ2ur the �roblem. 11 is imponm11 10 keep in mind 1ha1 yo11 con

ave only one more ,·ariable /hon )'OIi hare roles. You may J,m,e to make o subsriwtion Iha/ relates one .-oriable in terms of m1od1er.

5. lmplicitl) differentiate both sides of the equationwilh respeCJ to time.

6. SubS1itu1e all instantaneous rates and values of the, ariable and solve for !he rem!Uning rate or ,-ariable.

\J o\ v fotL,

�� \-'\�

�I,)� - '-\ -

� c�

':. -?> �1

1� cl\\ -

""IT - { J ft"\w_.

\J-= \i( "ih_ \J-:. ,r ( 4)

2.h

\J � \(o,r�

d\J \ 16 "tr • �:b

--

O,� - ��

-3- \ �ii �-

d�

!.h. - '3 -6:t \l."tr'

'-

, '-

A cone has a diame1er of 10 inches and a height of 15 inches. Water is beina poured into the cone so that

the height of the water level is changing at a ra1e f 1.2 inches per sec d. Al the iostam ,,•hen the radius

of the e,-'])Ose surface area of the water irz mc� al w t e 1s the volume of the water chan ?. "'

J:..- ..i.

(�fl (

% "'i:' - \ s g.h. <+=i) d�

, . ,uentify all of the variables involved in the problem. rocr�s

\.It�� ".). ,. ....... e... 2. Identify which. if any. of the variables in !heproblem that remain conslaOL

NOY\IL,

3. Identify the rare(s) that are given and !he ra1e tha1A- \.� \.>'\..1�you wish 10 find. dk-.t\J -

� � ·�! \Ste-

-;\� -.i. Write an equation. often a geometric formula or

\) = �"r�h. '("' - J..trigonometric equation. that relates all of the variablesh- 3in !he problem for which a rate is gi, en or for which a

'1� ! "'(i�J"h. rare is 10 be determined Subsritute any value Iha! ?, I'" = t"\ represents a variable that is consrant throughom the

\}-:. "i\\1f �4

·h r:. t "'-problem. !1 is impor1an110 keep i11 mind 1hat you can have olliy one more ,·ariable 1han you l,o,·e rates. l'2!!, mav 1,o,·e to molce a substi1111io111ha1 re/mes 011e ..L 11�3 va;,OD7e u, iernis q7 anotl1er. \} -:. ,., 5. lmplicill) differentia1e bolh sides oflhe equation cj\J ..L. �

2.�with respec1 10 1ime. -

C\ 1l' -ce -

d-+

!'1 - *1'('-l:i.(,.i) -

6. Substitute all insrantaneous rates and values of the d\J 4.i,r \A�\�variable and solve for !he remaining rate or variable. -._.-- -ctt

A ladder 25 feet long is leaning a2aiost the wall of a house. The base of the ladder is pulled away from the wall at a rate 0Q.fee1 per secog�

� "I..

a(fiow fast is th�of the ladder moving down the wall when the base of the ladder �from . the wall? � '2. � � -

44-_ _.;.,. 6,.'t )(. -t � = ,. ,; ,.'J(� -t ��" � 0 mm

n ........ 'X, 2

J('l)(2.) -', :2(2it) � = 0l8 + �� �

"\9 5: -lScit' �,..., ,�= :15

'ti z.� s,1o

� : 2-'i 5 - -.:i :: \-1. +c.d- \ dt - 4B ·'--l'l.. ___ �_....-_�_ ...... J

b. Consider the formed by the side of the house, the ladder, and the ground. Find the rate at which the ea of the trian is cban1rim? when the base of the ladder is 7 feet from the wall.- - 'IC:::::

A:. f �"'"' - ..L. �"' � � .l.x.. i � '1 � - -� �· -- d-b

iC. t- = i(:l'>(2.'+)-t i( ;)(-�) cA- - :iu. - � Tt - "I 2.'f

� � \�, .'{ 58 +t�{\le-\ c. Find the rate at which the angle formed by the ladder and the wall of the house is changing when the

base of the ladder is .2. feet from the wall. ? S,\'\e :. L

'2.S

\

0,'Jc �� An airpl e is flying a1 an @jtud�f 5 miles and passes directly o,·er a radar antenna When the plane is

IO mjl I the antenna. the rate at which the distance between the antenna and the plane is changing is

G!40 miles per hojib. What is the �e? - �

b: t

1 '�-' .2x �

et

.i(tb)� • i; .. '. , , .... �

l., , , dx _- -

, , � :a. 'l. dt ,, , 5 "T \0 = 'i

� �

�U).Ji�s

\'l.5: � ci�

The radius of a ��erefs��easing at a rate olQ. inches per mintiia Find the rE of change of the sillfacl';>

. �h, * tk,. ii:' cLA-area of the sphere when the radius �sr-

"l� 4k A ::. 41rr 'l..

t: '6irr�

� = t\\(,1(�) � -:. ' 9 b 11 U'\ 2. \(YU,�\

�

A spherical balloon is expanding at a rate of@1[ cubic inches pe�

nd. How€1 is the surface:�

the balloon expanding when the radius of the balloon is 4 inches.

\f-:. �nr'3 .Q;i_ -;. Ai:-n-r � & d,'t'

(ofrtr � 4rr( �')��

Date ________ Class ___ _

Day #32 & 33 Homework

1. The radius rand area A of a circle are related by the equation A = rrr. Write an equation that relates dA t1r A � di"° 10 7b. ':. 'n (

d. Pt -:. .2.-� ' k. (it ot

2. The radius rand surface area Sofa sphere are related by the equation S = 4 m2. Write an equationth I

dS tir 2. at re ates -;;; 10 dr .

S .., "tit" r

The radius r. height h. and volume V of a right circular cylinder are related by the equation 1 · = m2h. Use this relationship ro answer questions 3 - 5.

3. How is '! related to ! if h is constant?

\J :. ii l-\ '(' :I.

4. How is '! related to ';J; if r is constant?

(}..V : �iT'hf' �d.t d.t

\J :. iT" '\ -a. . Y\

il : '\'Ti"' '2. �d. t- d.\::

5. How is '! related to :!" and �; if neither r nor h is constant?

\l =- �'(' �. \-\

o.._1/ - :.

!:t 2\i ('" 21:: . "" -+

d.t

:l.,rt"h. �d.t

Use the follo,,ing information to solve problems 7 - 9.

Th�

ep�f a rec1ang.le is decreasing at lhe ra1e o 'l while lhe widlh is increasing at the rate

� emf When I= 12 cm and w = 5 cm. find each of the rates of change of each quantity indicated�low.

7. Area

G3 (\:. .l,w

l.·i �

8. Perimeter

1>: a � :2.w

�? .- . 4,-t

;2. il �

� clw

,._ �

!A- • i·w i �---- . 4.t �� :2.( -2..) -t �(�) ck - ':.

oA:: � - s(-2) + I 2. ( l-) -

6-Pr ': -10 + ).it

d_,-. : \ \4 <:.�.2· I �1

9. Lenglh of a diagonal of the rectangle" � .l,.

.l '\ ""' -:. �

di: d.w cu �J ... db. -t .l.w dt; = dX dt

:)(1�(-i)-,, �(�)(1-) :: 2()i) i - 'T t> -\- 2..0 - 21.o �-

d,\:; - d,8 - °'-�

�� c!.t

d..� (o : Cw... �

\�]

I 7-. -z. -t 5 2...� >-. '2..

I lo <\ -=-- .,.. J..

'J. -:: 13

r s ---

c£:L., A spherical balloon is inflated with helium at the rate c{\.OO;c ft3;;:> d..t­

d.C-I 0. How fast is the balloon· s ..:-.. ,,.. ·

when the radius is� r

\J� ± "(l'"� �!

c}..\J 'l. o.r - '"\trf" �

- -d.t

\001': 4\-rr( rs) 'l. �

\OOV -:. \0 0 TI' 4.Y' -�t'

.tr � \ i. +� � -

c�

.. mg

�\

11. How fast is the surface area increasing "henthe radiu s is 5 feet?

Pt.�

c;_A, - �cU: ell\

-- -

C..t'

d.� -

-

C,'t'

ir1T or ')..

""' cg\\.,.�

-Z-n (cs')(�

f4DT �1. \�j

"' - 1:2.

d.1/ -- -

d.t

\C-:

lS ir\,(. e"'l3 2 l:c

J .;- \i ( "i) -a. �'\�2. a,\,

IC-:. \OOTI"' � ��

,� :.. \ �'If

+t.t..1':. � �l)t(_\

au���w· 1'..o �ag�ainst the wall of a house. The base of the ladder is pulled away from the wall at a rate o eel r seco Use this information to complete exercises 13 - 15.

1� � 13. How t€f is the to�the ladder moving down the wall \\hen ils base i�'it-om the wall?sl.i. X�+ lo{�-:. :2-S

' ¢� l,J

, 2.5 ::2.x � "" 'l... � :: 0 ' ci.� � a.�'

', Z(.2.t)) t; -t 2.(1c;)( � :O

'-+--'j..,._� "rO � � bO

� 1L.,:_1: __ +_c..«.:_\._\ �->j: 14. 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 ladder is 9 feet from the wall.

A ..L. . 2.... c:kx - - 2.� �

,.,. =- � )<. lj ,. � - .J d.t

��)* = -2(0.'i..�o.A- - .l. �. '4 -t .l. X �cit - :). dt .J � dt

� , .l f-.l!. \( ci'' t �(�'v 2\dit" - � \�}\ ') ... . I}, /I

��t\<'.\.85l

.s\!.- -\1Qt- �

..,_1..+ q1-

:: 2.5� Y. 1.-:. s 4 � -

'f. -:. J" 5 � I.\15. Find the rate at which the angle berween the ladder and the ground is changing when the base of

the ]adder is 7 feet from rhe wall.

- �'"' e

2.

?--S

(

r "\S - li d.\1/ � � T- :,. !2.r ::. IS'-.

- �. � ----�� r:. ,.If\\ Water is flowin�te o 50 cubic meters r · ute from a concrete conical reservoir. The radius of · '·the reservoir is�d the height m ·se this information to complete exercises 16 and 17.

--"' 16. How fast is the water level falling when the water ief"mete�?

'J =- � -rr < 1 \.-. .i::!_ = 2. l. c;. ,. "' :lo !.b.\J -=- � 11 ( � �

') 1. \.i

a:t ""t d. �.! \.

- t:io :: �--.s ...-('S)i. �V:. \nl���� �· �" �

- 'g-: :l'l.S'IT �<ik-\J:.

17. How fast is the radius of the water"s surface changing when the water is 5 meters deep?

.?.r:: \� M. : -=- ,rr::a

kd.� ,., �t

h- .2o- r - 1'5

\· �'.l. , 1- -.....- i • - 1 'I - .3" ,s

\/-=-�Tl('� "'t5

2.t"" ":,. 1 ,

- rso -= �-; 1T ( �)1.

1;- so : � ,r ti!::.?- 4-�

_ - LtlO - -\_.....,.§-. -Y'Y\-�--f-_.-�-.-vtt.-:--i

·\ - � ,s-rr - L. _:.:IS1r:..!!----------

r:, 7'S/:,.. do/•� 18. A man 6 feet tall walks at a rate o@ pe second toward a street light that is 16 feet lllil. At

what@te is the len�f his shadow changing when h�om the base of the light?

\\,

1�t ij

\Ox. :: G:, �

lO J.� - (.. «l'!J� J.-l

\o � = b(-SJ'f.. x-+�

\� '1- '"" �)(.�le, J 10 x.-: (o

'j

� - -\--3-+-w"".'""A _p_v-�-�--,\ d.-t- .

19. lf the volume of a cube is increasing at a rate�

d each edge is increasing�

what is the length of each edge of the cube?

'\J = c?

4't � e,. '4

�::: \ ;2, lne,'-'e�

20. �ume of a cube is increasing at a rate oQ!_in'@i��I�� the surface area is increasing a1� what is the length of each edge of the cube?

� \J-:. e1 A= �e. '2..

�" 2 2 cl.e. o..Pr- _ l :2,(...

cle d.� = .;,C. c.� lit- - d.x

\'2.. - k\).c. - cl-t

3 e. 2.. - ?-"\' � -:::. 0

3c. (� - 'o)::. 0 � � ,��'gi.�\

21. On a morning when the sun will pass directly overhead. !he shadow of an 8�foot building onlevel ground is 60 feet Jong. Al !he mome01 in question, the angle B the sun makes with the groundis increasing at the rate of0.27 radians per minute. At what rate is the length of the shadow

�I

decreasing? � e s:

4o..-e ::

e·""' --- -- ___ ... ...