Related Rates Notes

RELAТED RAТES NOTES Objectives: • Find а related rate. • Use related rates to solve real-life proЫems. Finding Related Rates We have seen how the Chain Rule сап Ье used to find dy/dx implicitly. Another important use of the Chain Rule is to find the rates of change of two or more related variaЬles that are changing with respect to time. For example, when water is drained out of а conical tank (see Figure 2.33), the volume V, the radius r, and the height h of the water level аге all functions of time t. Volume is related to raoius and heighl Figure 2.33 ) clv- Knowing that these variaЫes аге related· Ьу the equation V = - r2!-i, you сап differentiate implicitly with respect to t 3 to obtain the related-rate equation: �6 \-"'2.t:>t:> U с.,i �ц ,- -:::; 1�-l, dv " c\-t:_ J,� J(Cf 2-)(.1)� + Cn)1.../,z., r � � - ----- 11.f .?-а� -з( Jk -+ J.:: z,.-� d< J. )f оте, From this equation, you сап see that the rate of change of V is related to the rates of change of both h and r. ExamRle 1: The variaЬles х an areJшtЬ...di erentiaЫe functions of t and аге related Ьу the equation у = х2 + 3. Find dy/dt when х = 1, given that dx/dt = 2 when х = 1. cr -;::: С{%-== c,{-f: - ? � � 'i."2- ...\-� х= \ d� Zx "Jt �� - 2-(1)@ =) )'( $[ 4 с z ______ , d-t:. ProЫem Solving with Related Rates Example 2: А реЬЫе is dropped into а calm pond, causing ripples in the form of ric circles, as shown in Figure 4. The radius r of the outer ripple is increasing at а constant rate of 1 foot per second. When the radius is 4 feet, at • at rate is the total area А of the disturbed water changing? . t�c.. -=.,., �t)\\)� \5, � �'(" t1,t..• ··1s-�c. ' t 4../ -т '"::::: л_ F'1)1.).D �i.s � =/ t:.�д.Nбl� dд dt:.. \)6е:.� � C.,\9-(:..,LJё. 'Df���Tt ir6 \f\.,\.\>\..,\,C,\j'-""'-/ (.J.)j (lbl>�C,\ � -t:., �о. t�. i'(" =- \ dt.. J } i-t- -=-=-=-=-=-=..:::.-=�----�======-= <=� :rд i;-1:2;'"'°' <7 о ,r 11. � 1 Total агеа increases as the outer radius lncreases. w\i�� � ,д� 0� 4 �Т Т� WA're.,Q. , '2..дD\l)sf ioure� .t> \.S'ТUgВt.D )!:» CUдt-=>�<; д.- А �1'� oF �1i �t-7-/� ' Guidelines for Solving Related-Rate ProЫems 1. Identif'y al1 given qпaпtities апd qllantities to Ье clereгmined. Make а sketch апd label tJ1e qнaлtities. 2. Write an equatioп iлvolviлg the vaiiaЫes wlюse 1·ates of change eitl1eг аге given or аге to Ье deteгmined. 3. Using the Chaiп Rнle, iшplicitly ditJeгentiate botl1 sides ot' tl1e eqL1atioп 11 1 it/1 гespect to time r. 4. AJte,· completing Step 3, sнbstitнte into tl1e resulti11g eqнation all kПО\\'П r values fог the variaЬles and t11eiг гates ot' cl1aлge. Tl1e11 solve fш the J геqнiгеd rate of change. -:'r-----����· The tаЫе below lists examples of mathematical models involving rates of change. For instance, the rate of change in the first example is the velocity of а саг. Matheшatical Model Verbal Stateшent The velocity of а car after traveling for 1 houг is 50 miles ре1· hollr. Water is being pшnped into а swimшing pool at а rate of I О снЬiс meteis ре1· hош. А gear is revolviпg at а гаtе of 25 i-evollltioпs рег шinute (1revulutioп = 27Т гati). х = distance traveled d x = 5 i/h \v hen t = 1 0ш l (f V = vo]шne of wateг in роо] dV - = 10ш3/h clt 8 = ш1gle of гevolнtion cle = 25(27Т) 1·ad/mi11 r/1 popLtlation of bacte1·ia is inc1·easing at а гаtе of 2000per hoLtr. = 11шпЬе1· in poplllation cLr; х - = 2000 bacreгia per hot1г dt Example 3: Air is being pumped into а spherical balloon at а rate of 4.5 cuЬic feet per minute. Find the rate of change of the radius when the radius is 1 foot. - f,r�з 2. )% (" 1Г(3 � =: : D\rye\2:Ьtv\1.д.Т6 l��L\c..tTL-i "'-'\ u:c6�� �() � (' .:::. \ � 1t,j :::. �1fl�. t Ч"б:: -г.15 = i+ = d= - � eдo1.us ,._ \ � 1\-\е 2-AD•� 1,s 1lG� fJiГA �т� e>F .y:,g, fio/мJN л '"" '//" ..J, 4--п-1\ � ) &{ r 4:r-... . =:,..J '2tzё J.::>D �;---""" w�t-) Note: The volume is increasing at а constant rate, ut the radius is not. As_ time incre�es, the radius is increasing more and more slowly. Example 4: An airplane is flying on а flight path that will take it directly over а radar tracking station. The distance s is :e,.LOC.. decreasin at а rate of 400 miles per hour when s=l0 miles. What is the s eed of the plane? rr � + (s, '2- "2. )(� i cli:., �� L ,..:> �,;. ID �� -400 �(:::, - d.�-dt: di cJ t: - = 2- � /.J roi' 1 t �-3>� , ' '' , ,' 5 - {�-3,(Q -l/2-6-z_S� JД I , , ":> '2..... !>� Ап airpl:1n� is Пyin� at un u!titшk ut· б miles, s 111iles from Lhe st,нinп. )5 � �� ..:.3(о ID "1ioc,-�(I" d-t:: (�ooj �ее.о-=- \Jc.-1..,C)C,(�'f \,\;)"" � ?t-A...:)6 1.S Р.'/ ID M\l.ЬS у fZ..DtV'.. 1� � "\"�� с.. U }.:)� Av:> �'\AT1Dl.) \'1� '5�� ,� f;OO м,(нt.