c) f,.. 1)D.() \ : 1,0>,~J4 I 'f:ij-' 0\.-

,,: D~ 2~ (018,\5

~ ~~ "q,IO,\\,\~

AP Calculus

Related Rates

. ..

Related Rates

1.) The radius "r" of a sphere is increasing at a rate of 2inches per minute. Find the rate of change of the volumewhen r = 6 inches and r = 24,iflches.dr _I")~ w. ~'dt- - c: "",n ~ ~f ':. Ydv I \ 7.. d",(" dv"'Jb ~ ~ -nr'C clt ~ (it

c1 i()'3(-= lP ~1ll~~) (7,.') = ~ ::& 21¢€>1T rv;=;-fl

r~2L..\ L.Mr(;24~)l~) -::.~~=-%og1r~~__. __ (t11

2.) Ripples in a Pond: A pebble is dropped into a calm pond,causing ripples in the form of concentric circles. Theradius "r" of the outer ripples is increasing at a constantrate of 1foot per second. When the radius is 4 feet, at

, --what rate is the total area "A" of the disturbed waterchanging? _ A .¢If'{ '2

d" - \~~ dPt. ' 9!:r:.\ - - - ~ om-'f" d..-t..~ s e c, d\:; a..

~. _ .~ ('-\)t,) : &r ~r~4-~~+ d.~ - O£'l ..' SaL.

3.) 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 3c:xn- -dl - ~e..c..

" ."

4.) Find the rate of change of the volume of a cone if therate of change of the radius is 2 inches per minute andh = Jr when r = 6 inches.

dv =.?dt .c:ir = cD)JI1:":<it m)r,h=or

~,

v = ~"fT"(' z.hV:: ~1T' yo" ('0(')

'!tV -= 1)'<d.v 2. ci~cL-t, ~ 3~(' "~

dv (, _ ,In'dt : ~~('=bI..t:J)~') - ~\lon- ;;;'1')

,/~

(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 thedepth of the water when the water is 6 feet deep?

d\J. = 5-!!.:dt ~\'"

dhdX ~ '?\j: ~"t\(" zh.,

:~\ \J, ••• a.. t.,,-;. ~h2. h-~' ~ 'Q e;;

,_.\ 'Io ••• ~.

J:._Jeh - 10

~'" ': \0 r-_.r i-±-h

5

• •

; -

A rocket, rising vertically ~is tracked by a radar stationthat is on the ground 3000 feet from the launching pad.At what rate is the angle of elevation changing when therocket is 4000 feet up and rising vertically at 5000 feetper second? dh ~+J

dt :::sCOO' I!::"e.~

h~40()D

~'3000 -rf\ to...) e :::.'300-t:>

<3e,c 2.. G c.:i§ -:. ...1... d h 'd\:. aooo ci.-1:. '

d& S qg .~c. 2. ~q?1~ n:::~w.·

.4:~~}- .~..~...~.Ql.dk~c.7.) Air is blown into a ~herical balloon. How fast is the

radius changing whenthe air is blown into the balloon atthe rate of 10 centimeters per second at the instant the

. radius is 5 centimeters?' , -

dr _?d-t - ..2:';L _ ID~1I

dt - -se "'"'. r:: is CX'Y'"

\/ ~~ r~ct.\J 'a- ~---, ": y.~ ~dJc - - - -. --- .}b ;: 4 -rr ('2.-s) ~~~~_ cLrIO'Tl ?f.'- -cc-~

7sV A weather balloon is rising up from the ground at theL / rate of 140 feet per minute. This. balloon is being tracked

by radar located on the ground 500 feet from the point oflift-off. Find the rate of change of the angle of elevationwhen the balloon is 500 feet above the ground.

tin ~~ =\4(:) ~

"" de-~ ~.~

-'k :=.. So (:)av1- n :=: f:oo @ -=- '1fJUt o« L\S0

: h~Ct,n 9 :::~

~"A ~,~ ~ -L <1h de' \ ( ". <Xl- \::7 G-t -:.~on -d.-!b Se.c..2.(:?~) (;;u-) =- Soo \~(),

9.) 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 perhour when s'= 10 miles, what is the speed of the plane?

- - ==::> d~ _-400ml~dt - ho\..U""

lLJ

t::: \()('()\\eA--

0, •

,,'

10.) A television camera at ground level-is filming the lift-offof a space shuttle that is rising vertically according to theposition equation, s =50t2, where "s" is measured in feetand "t" is measured in sec.onds.The camera is 2000 feetfrom the .launching pad. Find the rate of change in theangle of elevation of the camera at 10 seconds after lift -off. t 1':- .o

~'Q.t ~+-a.n.& c a~eh

l1 o ,S ~g~ _ ~- ) c'~c.. d-b - [}..o:t. ' ;

c:b "/1, \ n) ~ - \ ,~(, \,11', d-t. - 8 ',:-'d%t ~ o. c-~q rd~ec...

11.) A trough is-12 feet long and 3 feet across. It ends areisosceles triangles with altitudes' of 3 feet.

a.) If water is being pumped intothetrough at 1 cubic.feet per minute" how--raSt-is the water level 'risingwhen "h" ~s1 foot deep? j;!:- \wd 1%'J ~ {( cU6)'rv '* . ~'I':: \ (cl b) \~ 3 ::\~(\) clb, ~~/~, _ dt!\J ~ (p(a b,) L:\- trotA - d~

b) If"-h~'to dl.. .. f~li·. t e water IS nsmg at a rate 0 ~r mlnute3

when h = 2, determine the rate at which wa~er is gitK.hb · d · h. gh \"2..,,,elng pumpe Into t ,e trou . - = 'H

a" ,'.' ~ ,~ ~~ ~1'9,.d-~~~Jt; ~-l;, <

~ d2 tOl),cti\,.ct \:- ,'/,.' Jdv 3, ()~3J"db = 1,,\"1' tir« {\

ifIG) A ladder 25 feet long is leaning against the wall of al7 house. The base of the ladder is pulled away from the

wall at a rate o(~t-per second.a.) 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~)~ dlW)( ~)-:. 0~t . (Po + ~'O ~/.L:t = 0

X '7. + 'j'Z.~2S 2 ~/di ':..- 3/2

-Pt-Is<!<-I") cl-lC. c!~ 0()(~~ t dJ ~ ': ..~~ "~l6)4)l~) + ..:2C1x.. 7d~ ':.. 0d (,)(~) ~ ;zOix. <ll.t- ) ~ 0 . . ~ I _ If 8) .-PH

~ _ -J. ~J .' uc : (1' Is~(..,dt - 1'2. t.i,c.Ib.)J Consider the triangle formed by the side of the house, theV 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 I '61'5:.-~J n~'~~~ )'J;.-=- ~ x : Q..t "'--AI

7-. . '-- / d l- d 1- ~.,.y~", ~ '>-~:s: .~I, ~,=~~~~x"')\~~ f"(:,,:..,..,~_ft-F ~".,; ::' ~;2S -"i. . u( I r' " 'f. /) '-\~ ..~ J~ 't- -'i- ({-lS~'- Y~) '- ~;~L~~ ~) 7~~ ..,j ~ .•<i~:~:~

.' - - ~[4~-~1~1 q(o.f.tJerc,c,c.) Find the rate at which the angle between die ladder and .

the wall of the house is changing when the base of theladder is 7 feet from the wall.

:L-5lrv f} -= 1---)

eke- ) ~(jQ 5 ~ ~ ~ ;-?' ;~~:/\ ~ e-. ~ _' .-L ('2.} ._.~J \j dt - .',,'Z-S" '.

de ,cadi .-at ~~ -~ft;;,~~··· "

. .

- ~.;

13.) The radius "r" of a sphere is increasing at a constant rateof 0.04 centimeters per secon-a~---- ..

a.) At the time when the radius of the sphere is 10centimeters, what is the rate of increase of itsvolume?

V==- ~ 11' $, .

~ ::: 111i"Y z ~

dv ("M. = 4-cr \ 0 0) o·0,"+~ = \ lo1r CJY'/ see:

b.) At the timewhen the volume of the sphere is 361rcubic centimeters, what is the rate of increase of thearea of a cross section through the center of thesphere? . u &..l' . '2.. 3loTT :::: '3 '1fY

..~..C\ =1TY ,\ 3<".' ~. 3<..0 ~3 '<"'

d A- ~:;hJY<l!: (()"7I:::: v ~db - d~ cA"

~A 3~~~ ::; <.a1l (o.ou.)

~,.. 0 I q,\{'fj" -'> ~?Jcvtlse.c.c.) At the time when the volume and the radius of the

sphere are increasing at the same numerical rate,what is the radius?

,dv ,·L c:U: ,\ rc~ r. ·a-t·, '. .:.~'.a; G;~:' Oll41rf 2.~~)' =, 0,:. "1

41i('~o,d4)--~0 1~~

'1- \., ,~ ,--::;.,\ . ) ~ '" 3 5\1. A M,~~~-~ ':..'r= B~ IV • , (...A'\

. ..•" . ,

14.) Sand is falling off a conveyor onto a conical pile at therate of 15 cubic feet per minute. The diameter of the baseof the cone -isapproximately twice the altitude. At whatrate is the height of the pile changing when it is 10 feet---------high? 4Y,.. -ft-.3,.

dt - \S- ~

\ \_~ \...:..V~ ~ 1\,n n

,,1:- ~ ll'h3

d" '1rh' sib~ :: clt

cl-'"\S -:::.."U'"' \ 0 0 ~

~ ~-TJ. ~ <!b..~o~ I~ 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 is 25- r--------

feet above the ground -

:~~&h~,-d.e- -::::?c:L~

"'- ~ "L S-'

-,

..,.:;..'