Find a related rate. Use related rates to solve real-life problems. Finding Related Rates

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1 8 Chapter Differentiation.6 Related Rate r Find a related rate. Ue related rate to olve real-life problem. r r h h Finding Related Rate You have een how the Chain Rule can be ued to find d implicitl. Another important ue of the Chain Rule i to find the rate of change of two or more related variable that are changing with repect to time. For eample, water i ained out of a conical tank (ee Figure.), the volume V, the radiu r, and the height h of the water level are all function of time t. Knowing that thee variable are related b the equation V r h Original equation ou can differentiate implicitl with repect to t to obtain the related-rate equation d V d r h dv Differentiate with repect to t. r dh h r r dh rh. From thi equation, ou can ee that the rate of change of V i related to the rate of change of both h and r. h Eploration Finding a Related Rate In the conical tank hown in Figure., the height of the water level i changing at a rate of 0. foot per minute and the radiu i changing at a rate of 0. foot per minute. What i the rate of change in the volume the radiu i r foot and the height i h feet? Doe the rate of change in the volume depend on the value of r and h? Eplain. Volume i related to radiu and height. Figure. FOR FURTHER INFORMATION To learn more about the hitor of related-rate problem, ee the article The Lengthening Shadow: The Stor of Related Rate b Bill Autin, Don Barr, and David Berman in Mathematic Magazine. To view thi article, go to MathArticle.com. Two Rate That Are Related The variable and are both differentiable function of t and are related b the equation. Find, given that d. Solution Uing the Chain Rule, ou can differentiate both ide of the equation with repect to t. d d d When and d, ou have. Write original equation. Differentiate with repect to t. Chain Rule deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

2 Problem Solving with Related Rate In Eample, ou were given an equation that related the variable and and were aked to find the rate of change of. Equation: Find: d In each of the remaining eample in thi ection, ou mut create a mathematical model from a verbal decription. Ripple in a Pond.6 Related Rate 9 Total area increae a the outer radiu increae. Figure. A pebble i opped into a calm pond, cauing ripple in the form of concentric circle, a hown in Figure.. The radiu r of the outer ripple i increaing at a contant rate of foot per econd. When the radiu i feet, at what rate i the total area A of the diturbed water changing? Solution The variable r and A are related b A r. The rate of change of the radiu r i. Equation: Find: A r da With thi information, ou can proceed a in Eample. d A d r da r r 8 quare feet per econd Differentiate with repect to t. Chain Rule When the radiu i feet, the area i changing at a rate of Subtitute for r and for. Simplif. 8 quare feet per econd. REMARK When uing thee guideline, be ure ou perform Step before Step. Subtituting the known value of the variable before differentiating will produce an inappropriate derivative. GUIDELINES FOR SOLVING RELATED-RATE PROBLEMS. Identif all given quantitie and quantitie to be determined. Make a ketch and label the quantitie.. Write an equation involving the variable whoe rate of change either are given or are to be determined.. Uing the Chain Rule, implicitl differentiate both ide of the equation with repect to time t.. After completing Step, ubtitute into the reulting equation all known value for the variable and their rate of change. Then olve for the required rate of change. Ru Bihop/Alam deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

3 50 Chapter Differentiation The table below lit eample of mathematical model involving rate of change. For intance, the rate of change in the firt eample i the velocit of a car. Verbal Statement Mathematical Model The velocit of a car after traveling for ditance traveled hour i 50 mile per hour. d 50 mih t Water i being pumped into a wimming pool at a rate of 0 cubic meter per hour. V volume of water in pool dv 0 m h angle of revolution A gear i revolving at a rate of 5 revolution per minute revolution rad.d 5 radmin A population of bacteria i increaing at a rate of 000 per hour. number in population d 000 bacteria per hour An Inflating Balloon Air i being pumped into a pherical balloon (ee Figure.5) at a rate of.5 cubic feet per minute. Find the rate of change of the radiu the radiu i feet. Solution Let V be the volume of the balloon, and let r be it radiu. Becaue the volume i increaing at a rate of.5 cubic feet per minute, ou know that at time t the rate of change of the volume i dv 9. So, the problem can be tated a hown. Find: dv 9 (contant rate) r To find the rate of change of the radiu, ou mut find an equation that relate the radiu r to the volume V. Equation: V r Volume of a phere Inflating a balloon Figure.5 Differentiating both ide of the equation with repect to t produce dv r r dv. Differentiate with repect to t. Solve for Finall, r, the rate of change of the radiu i foot per minute. In Eample, note that the volume i increaing at a contant rate, but the radiu i increaing at a variable rate. Jut becaue two rate are related doe not mean that the are proportional. In thi particular cae, the radiu i growing more and more lowl a t increae. Do ou ee wh? deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

4 .6 Related Rate 5 The Speed of an Airplane Tracked b Radar See LaronCalculu.com for an interactive verion of thi tpe of eample. 6 mi Not awn to cale An airplane i fling at an altitude of 6 mile, mile from the tation. Figure.6 An airplane i fling on a flight path that will take it directl over a radar tracking tation, a hown in Figure.6. The ditance i decreaing at a rate of 00 mile per hour 0 mile. What i the peed of the plane? Solution Let be the horizontal ditance from the tation, a hown in Figure.6. Notice that 0, Find: d 0 and You can find the velocit of the plane a hown. Equation: d 00 6 d d d d mile per hour 8 Pthagorean Theorem Differentiate with repect to t. Solve for d Subtitute for,, and. Simplif. Becaue the velocit i 500 mile per hour, the peed i 500 mile per hour. d. REMARK The velocit in Eample i negative becaue repreent a ditance that i decreaing. A Changing Angle of Elevation tan = ft Not awn to cale A televiion camera at ground level i filming the lift-off of a rocket that i riing verticall according to the poition equation 50t, where i meaured in feet and t i meaured in econd. The camera i 000 feet from the launch pad. Figure.7 Find the rate of change in the angle of elevation of the camera hown in Figure.7 at 0 econd after lift-off. Solution Let be the angle of elevation, a hown in Figure.7. When t 0, the height of the rocket i 50t feet. d 00t velocit of rocket Find: d t 0 and Uing Figure.7, ou can relate and b the equation tan 000. Equation: tan See Figure When t 0 and 5000, ou have d ec d 000 d d co 00t t 000 radian per econd. So, t 0, i changing at a rate of radian per econd Differentiate with repect to t. Subtitute 00t d for. co deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

5 5 Chapter Differentiation The Velocit of a Piton In the engine hown in Figure.8, a 7-inch connecting rod i fatened to a crank of radiu inche. The crankhaft rotate counterclockwie at a contant rate of 00 revolution per minute. Find the velocit of the piton. Crankhaft Piton Spark plug 7 Connecting rod The velocit of a piton i related to the angle of the crankhaft. Figure.8 a b c Law of Coine: b a c ac co Figure.9 Solution Label the ditance a hown in Figure.8. Becaue a complete revolution correpond to radian, it follow that d radian per minute. Find: (contant rate) You can ue the Law of Coine (ee Figure.9) to find an equation that relate and. Equation: When, d 00 d 6 co d 6 in ou can olve for a hown. 7 co Chooe poitive olution. So, 8 and the velocit of the piton i d , 08 inche per minute. 7 co 0 d 6 in d co d d d 6 in 6 co d REMARK The velocit in Eample 6 i negative becaue repreent a ditance that i decreaing. deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

6 .6 Related Rate 5.6 Eercie See CalcChat.com for tutorial help and worked-out olution to odd-numbered eercie. Uing Related Rate In Eercie, aume that and are both differentiable function of t and find the required value of / and d/. Equation Find Given. (a) (b). 5 (a) (b). (a) 8 (b). 5 (a), (b) Moving Point In Eercie 5 8, a point i moving along the graph of the given function at the rate d/. Find / for the given value of. 5. d ; centimeter per econd (a) (b) 0 (c) 6. d inche per econd ; 6 (a) (b) 0 (c) d 7. tan ; feet per econd (a) (b) (c) 0 d 8. co ; centimeter per econd d 5 d d d, (a) (b) (c) 6 WRITING ABOUT CONCEPTS 9. Related Rate Conider the linear function a b. d d d 0 6 d 8 If change at a contant rate, doe change at a contant rate? If o, doe it change at the ame rate a? Eplain. 0. Related Rate In our own word, tate the guideline for olving related-rate problem.. Area The radiu r of a circle i increaing at a rate of centimeter per minute. Find the rate of change of the area (a) r 8 centimeter and (b) r centimeter.. Area The included angle of the two ide of contant equal length of an iocele triangle i. (a) Show that the area of the triangle i given b A in. (b) The angle i increaing at the rate of radian per minute. Find the rate of change of the area and. (c) Eplain wh the rate of change of the area of the triangle i not contant even though d i contant.. Volume The radiu r of a phere i increaing at a rate of inche per minute. (a) Find the rate of change of the volume r 9 inche and r 6 inche. (b) Eplain wh the rate of change of the volume of the phere i not contant even though i contant.. Volume A pherical balloon i inflated with ga at the rate of 800 cubic centimeter per minute. How fat i the radiu of the balloon increaing at the intant the radiu i (a) 0 centimeter and (b) 60 centimeter? 5. Volume All edge of a cube are epanding at a rate of 6 centimeter per econd. How fat i the volume changing each edge i (a) centimeter and (b) 0 centimeter? 6. Surface Area All edge of a cube are epanding at a rate of 6 centimeter per econd. How fat i the urface area changing each edge i (a) centimeter and (b) 0 centimeter? 7. Volume At a and and gravel plant, and i falling off a conveor and onto a conical pile at a rate of 0 cubic feet per minute. The diameter of the bae of the cone i approimatel three time the altitude. At what rate i the height of the pile changing the pile i 5 feet high? (Hint: The formula for the volume of a cone i V r h.) 8. Depth A conical tank (with verte down) i 0 feet acro the top and feet deep. Water i flowing into the tank at a rate of 0 cubic feet per minute. Find the rate of change of the depth of the water the water i 8 feet deep. 9. Depth A wimming pool i meter long, 6 meter wide, meter deep at the hallow end, and meter deep at the deep end (ee figure). Water i being pumped into the pool at cubic meter per minute, and there i meter of water at the deep end. m m min m 6 m (a) What percent of the pool i filled? (b) At what rate i the water level riing? m 6 deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

7 5 Chapter Differentiation 0. Depth A trough i feet long and feet acro the top (ee figure). It end are iocele triangle with altitude of feet. ft min. Contruction A winch at the top of a -meter building pull a pipe of the ame length to a vertical poition, a hown in the figure. The winch pull in rope at a rate of 0. meter per econd. Find the rate of vertical change and the rate of horizontal change at the end of the pipe 6. ft ft h ft ft d = 0. m ec (, ) ft ft 9 6 (a) Water i being pumped into the trough at cubic feet per minute. How fat i the water level riing the depth h i foot?. Moving Ladder A ladder 5 feet long i leaning againt the wall of a houe (ee figure). The bae of the ladder i pulled awa from the wall at a rate of feet per econd. m (a) The winch pull in rope at a rate of feet per econd. Determine the peed of the boat there i feet of rope out. What happen to the peed of the boat a it get cloer to the dock? 5. Air Traffic Control An air traffic controller pot two plane at the ame altitude converging on a point a the fl at right angle to each other (ee figure). One plane i 5 mile from the point moving at 50 mile per hour. The other plane i 00 mile from the point moving at 600 mile per hour. (a) At what rate i the ditance between the plane decreaing? (b) How much time doe the air traffic controller have to get one of the plane on a different flight path? 5 ft 5m ft ec Figure for (b) Suppoe the boat i moving at a contant rate of feet per econd. Determine the peed at which the winch pull in rope there i a total of feet of rope out. What happen to the peed at which the winch pull in rope a the boat get cloer to the dock? Figure for FOR FURTHER INFORMATION For more information on the mathematic of moving ladder, ee the article The Falling Ladder Parado b Paul Scholten and Anew Simoon in The College Mathematic Journal. To view thi article, go to MathArticle.com.. Contruction A contruction worker pull a five-meter plank up the ide of a building under contruction b mean of a rope tied to one end of the plank (ee figure). Aume the oppoite end of the plank follow a path perpendicular to the wall of the building and the worker pull the rope at a rate of 0.5 meter per econd. How fat i the end of the plank liding along the ground it i.5 meter from the wall of the building? Ditance (in mile) r Figure for. Boating A boat i pulled into a dock b mean of a winch feet above the deck of the boat (ee figure). (b) Conider the triangle formed b the ide of the houe, the ladder, and the ground. Find the rate at which the area of the triangle i changing the bae of the ladder i 7 feet from the wall. 0.5 ec 6 Figure for (a) How fat i the top of the ladder moving down the wall it bae i 7 feet, 5 feet, and feet from the wall? (c) Find the rate at which the angle between the ladder and the wall of the houe i changing the bae of the ladder i 7 feet from the wall. Not awn to cale 8 (b) The water i riing at a rate of inch per minute h. Determine the rate at which water i being pumped into the trough. m mi Not awn to cale 00 Ditance (in mile) Figure for 5 Figure for 6 6. Air Traffic Control An airplane i fling at an altitude of 5 mile and pae directl over a radar antenna (ee figure). When the plane i 0 mile awa 共 0兲, the radar detect that the ditance i changing at a rate of 0 mile per hour. What i the peed of the plane? deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

8 .6 Related Rate Sport A baeball diamond ha the hape of a quare with ide 90 feet long (ee figure). A plaer running from econd bae to third bae at a peed of 5 feet per econd i 0 feet from third bae. At what rate i the plaer ditance from home plate changing? Figure for 7 and 8 Figure for 9 8. Sport For the baeball diamond in Eercie 7, uppoe the plaer i running from firt bae to econd bae at a peed of 5 feet per econd. Find the rate at which the ditance from home plate i changing the plaer i 0 feet from econd bae. 9. Shadow Length A man 6 feet tall walk at a rate of 5 feet per econd awa from a light that i 5 feet above the ground (ee figure). (a) When he i 0 feet from the bae of the light, at what rate i the tip of hi hadow moving? (b) When he i 0 feet from the bae of the light, at what rate i the length of hi hadow changing? 0. Shadow Length Repeat Eercie 9 for a man 6 feet tall walking at a rate of 5 feet per econd toward a light that i 0 feet above the ground (ee figure). 0 6 rd 8 90 ft nd Home t (0, ) m (, 0). Evaporation A a pherical rainop fall, it reache a laer of air and begin to evaporate at a rate that i proportional to it urface area S r. Show that the radiu of the rainop decreae at a contant rate.. HOW DO YOU SEE IT? Uing the graph of f, (a) determine whether i poitive or negative given that d i negative, and (b) determine whether d i poitive or negative given that i poitive. (i) f 5. Electricit The combined electrical reitance R of two reitor and R, connected in parallel, i given b R R R R where R, R, and R are meaured in ohm. R and R are increaing at rate of and.5 ohm per econd, repectivel. At what rate i R changing R 50 ohm and R 75 ohm? 6. Adiabatic Epanion When a certain polatomic ga undergoe adiabatic epanion, it preure p and volume V atif the equation pv. k, where k i a contant. Find the relationhip between the related rate dp and dv. 7. Roadwa Deign Car on a certain roadwa travel on a circular arc of radiu r. In order not to rel on friction alone to overcome the centrifugal force, the road i banked at an angle of magnitude from the horizontal (ee figure). The banking angle mut atif the equation rg tan v, where v i the velocit of the car and g feet per econd per econd i the acceleration due to gravit. Find the relationhip between the related rate dv and d. (ii) 6 5 f Figure for 0 Figure for. Machine Deign The endpoint of a movable rod of length meter have coordinate, 0 and 0, (ee figure). The poition of the end on the -ai i t in t 6 where t i the time in econd. (a) Find the time of one complete ccle of the rod. (b) What i the lowet point reached b the end of the rod on the -ai? (c) Find the peed of the -ai endpoint the -ai endpoint i, 0.. Machine Deign Repeat Eercie for a poition function of t 5 in t. Ue the point for part (c). 0, 0 8. Angle of Elevation A balloon rie at a rate of meter per econd from a point on the ground 50 meter from an oberver. Find the rate of change of the angle of elevation of the balloon from the oberver the balloon i 50 meter above the ground. r deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

9 56 Chapter Differentiation 9. Angle of Elevation A fih i reeled in at a rate of foot per econd from a point 0 feet above the water (ee figure). At what rate i the angle between the line and the water changing there i a total of 5 feet of line from the end of the rod to the water? (0, 50) Figure for 9 Figure for 0 0. Angle of Elevation An airplane flie at an altitude of 5 mile toward a point directl over an oberver (ee figure). The peed of the plane i 600 mile per hour. Find the rate at which the angle of elevation i changing the angle i (a) (b) and (c) 0,. Linear v. Angular Speed A patrol car i parked 50 feet from a long warehoue (ee figure). The revolving light on top of the car turn at a rate of 0 revolution per minute. How fat i the light beam moving along the wall the beam make angle of (a) (b) and (c) with the perpendicular line from the light to the wall? 50 ft 0 ft Figure for Figure for. Linear v. Angular Speed A wheel of radiu 0 centimeter revolve at a rate of 0 revolution per econd. A dot i painted at a point P on the rim of the wheel (ee figure). (a) Find d a a function of. (b) Ue a graphing utilit to graph the function in part (a). (c) When i the abolute value of the rate of change of greatet? When i it leat? (d) Find d and 60, 0, 0. Flight Control An airplane i fling in till air with an airpeed of 75 mile per hour. The plane i climbing at an angle of 8. Find the rate at which it i gaining altitude.. Securit Camera A ecurit camera i centered 50 feet above a 00-foot hallwa (ee figure). It i eaiet to deign the camera with a contant angular rate of rotation, but thi reult in recording the image of the urveillance area at a variable rate. So, it i deirable to deign a tem with a variable rate of rotation and a contant rate of movement of the canning beam along the hallwa. Find a model for the variable rate of rotation feet per econd. d 60, 75. P cm 5 mi Not awn to cale Figure for 5. Think About It Decribe the relationhip between the rate of change of and the rate of change of in each epreion. Aume all variable and derivative are poitive. (a) d (b) 00 ft d L, Acceleration In Eercie 6 and 7, find the acceleration of the pecified object. (Hint: Recall that if a variable i changing at a contant rate, it acceleration i zero.) 6. Find the acceleration of the top of the ladder decribed in Eercie the bae of the ladder i 7 feet from the wall. 7. Find the acceleration of the boat in Eercie (a) there i a total of feet of rope out. 8. Modeling Data The table how the number (in million) of ingle women (never married) and married women m in the civilian work force in the United State for the ear 00 through 00. (Source: U.S. Bureau of Labor Statitic) Year m Year m L (a) Ue the regreion capabilitie of a graphing utilit to find a model of the form m a b c d for the data, where t i the time in ear, with t correponding to 00. (b) Find dm. Then ue the model to etimate dm for t 7 it i predicted that the number of ingle women in the work force will increae at the rate of 0.75 million per ear. 9. Moving Shadow A ball i opped from a height of 0 meter, meter awa from 0 m the top of a 0-meter lamppot (ee figure). Shadow The ball hadow, m caued b the light at the top of the lamppot, i moving along the level ground. How fat i the hadow moving econd after the ball i releaed? (Submitted b Denni Gittinger, St. Philip College, San Antonio, TX) deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.

10 Anwer to Odd-Numbered Eercie A5 Section.6 (page 5). (a) (b) 0. (a) 5 8 (b) 5. (a) 8 cmec (b) 0 cmec (c) 8 cmec 7. (a) ftec (b) 6 ftec (c) ftec 9. In a linear function, if change at a contant rate, o doe. However, unle a, doe not change at the ame rate a.. (a) (b). (a) (b) If i contant, dv i proportional to r. 5. (a) 7 cm ec (b) 800 cm ec ftmin 9. (a).5% (b). (a) 7 8 mmin ftec; ftec; 7 ftec 57 (b) ft ec (c) radec. Rate of vertical change: 5 mec Rate of horizontal change: 5 mec 5. (a) 750 mih (b) 0 min ftec (a) ftec (b) ftec. (a) ec (b) m (c) 50 mec. Evaporation rate proportional to S dv kr 6 cm min 56 cm min 97 in. min; 5,55 in. min So k ohm/ec 7. dv 6r v dv 6r co v ec d, radec 55. (a) ftec (b) (c) About. About mih 5. (a) d mean that change three time a fat a change. (b) change lowl 0 or L. change more rapidl i near the middle of the interval. V r dv r ftec 7. ftec ftec 9. About mec d deemed that an uppreed content doe not materiall affect the overall learning eperience. Cengage Learning reerve the right to remove additional content at an time if ubequent right retriction require it.