Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm begins t slip ut. Assume that the bttm f the ladder slides away frm the wall at a cnstant rate f 2 ft/s, as represented by the arrw n each drawing. At any mment, the bttm f the ladder is x feet frm the wall, and the tp f the ladder is y feet frm the grund, as shwn in the center diagram. Fcus n the questin: What is the velcity f the tp f the ladder when it is a given distance abve the grund? as yu wrk thrugh the first few pages f this activity. Cnstruct Yur Understanding Questins (t d in class) 1. Label the distances x and y in the first and last drawings in Mdel 1. 2. When a. y = 8 ft, what is x? b. y = ft, what is x? Write dwn any equatins yu used t find yur answers. x 2 ft/s 2 ft/s 2 ft/s. Suppse we want t knw the velcity f the tp f the ladder when y = 8 feet. A gd way t begin answering this questin is t take an inventry f what is knwn and unknwn, assigning variables t each. D this by cmpleting the table belw. Descriptin at the mment f interest: (when y = 8 ft) Symbl r Variable Value If unknwn, write? Distance f the tp f the ladder frm the grund y 8 feet Distance f the bttm f the ladder frm the wall Velcity f the bttm f the ladder, which is the rate f change in x r y [circle ne] with respect t time. Units Velcity f the tp f the ladder, which is the rate f change in x r y [circle ne] with respect t time.
152 Differentiatin Applicatins 1: Related Rates 4. (Check yur wrk) There shuld be nly ne unknwn in the table in Mdel 1, and this shuld crrespnd t the questin (in bld) in Mdel 1. If yu listed a value f? fr bth velcities, read Mdel 1 mre carefully and change ne f these t an actual value. 5. Nte that x and y in Mdel 1 are changing with time ( t ), s each is a functin f t. a. Cnstruct a descriptin f what these symbls mean in the cntext f Mdel 1. i. dx ii. dy b. (Check yur wrk) If yu have nt already dne s, circle x r y in the last tw rws n the table in Questin. Write dx and dy in the apprpriate bxes f the table, and check that yur descriptins abve match the descriptins in the table. c. Is dy psitive r negative? Circle ne, and explain yur reasning. d. (Check yur wrk) Is yur answer t part c cnsistent with the fact that dx ppsite signs in this prblem? and dy have 6. (Check yur wrk) Did yu cite the equatin x y 100 in Questin 2? 2 2 a. If nt, g back and check yur wrk. b. Starting frm this equatin, use implicit differentiatin t generate an expressin fr dy, in terms f the variables and rates f change n the table in Mdel 1. (If yu are stuck, mve nt parts c-e n the next page.)
Differentiatin Applicatins 1: Related Rates 15 c. (Check yur wrk) Which is the derivative f the term dy dx 2x 2x 2x dx dy dx dx 2 x with respect t t? 2 dy x dx Circle ne and state what differentiatin rule is being applied here. 2 dx x d. Using this same differentiatin rule, find the derivative with respect t t f the term 2 y. e. Use yur answers t parts c and d, abve, t cnstruct an answer t part b. Check the resulting expressin fr dy with anther grup. 7. Use what yu have learned s far in this activity t find the velcity f the tp f the ladder in Mdel 1 a. When it is 8 feet abve the grund? Draw a picture and shw yur wrk. b. When it is feet abve the grund? Draw a picture and shw yur wrk.
154 Differentiatin Applicatins 1: Related Rates 8. Helium is leaking ut f a spherical weather balln at a cnstant rate f cubic feet per secnd. What is the rate f change in the radius f the balln at the mment when the vlume f the balln is.5 ft? It is helpful t draw a picture and yu may summarize the infrmatin in a table like the ne in Mdel 1. If yu are stuck, try the next questin, then cme back t cmplete 4 this questin. Nte that the vlume f a sphere is given by V r Descriptin at the mment f interest: when Symbl r Variable Value If unknwn, write? Units 9. (Check yur wrk) The previus questin can be slved by answering each f the fllwing questins. If yu skipped any f these, answer the questin, then g back cnsider hw it applies t the previus questin. a. What defines the mment f interest fr this prblem? (e.g., in Mdel 1, the mment f interest was defined by the time at which the tp f the ladder was 8 ft. frm the grund, r y = 8 ft.) b. What variables and rates f change are invlved in this prblem? Write a descriptin and assign a symbl and units t each ne. (Enter these n the table.)
Differentiatin Applicatins 1: Related Rates 155 c. Write dwn an equatin relating the variables in the prblem. d. Cnsider the equatin, y k x, where k is a cnstant and the variables x and y are each a functin f time, t. i. Use implicit differentiatin t generate a new equatin cntaining the rate f change f x with respect t t (and ther quantities), then slve fr this rate. Shw yur wrk. ii. Explain hw this relates t the weather balln prblem. e. What is the radius when the balln vlume is.5 ft? f. What is the sign f dv? g. What is the rate f change in the radius f the balln at the mment when the vlume f the balln is.5 ft? 10. (Check yur wrk) The fllwing is a checklist f steps that culd be useful in slving a related rates prblem. Which steps did yu use in slving Questin 8? Which steps did yu skip? Put a check mark next t the nes yu used. Draw a picture and assign symbls t relevant quantities. Summarize the infrmatin prvided in the prblem (e.g., make a table): Nte which quantities are changing ver time, and use crrect ntatin t express their rate f change with respect t time. Write a descriptin f what each variable and rate f change means, being mindful f units f measure. Identify which quantities are knwn and which are unknwn. Write dwn an equatin relating the variables yu identified. If pssible, use gemetry t eliminate ne r mre variables. Use implicit differentiatin t generate a new equatin relating the variables and their rates f change, then slve fr the unknwn. Answer the questin asked in the prblem remembering t state the units f measure.
156 Differentiatin Applicatins 1: Related Rates Extend Yur Understanding Questins (t d in r ut f class) 11. Tw straight rads intersect in the middle f twn. One runs nrth-suth, the ther east-west. Nah is driving nrth frm the intersectin at 20 miles per hur. Ellie is driving east frm the intersectin at 0 miles per hur. When Nah is 0. miles frm the intersectin and Ellie is 0.4 miles frm the intersectin, hw fast are they mving apart? (Try nt t lk back at yur checklist unless yu are stuck.) 12. The cnical water tank shwn at right is filling at a rate f 1 m (1000 L) per hur. 2meters At what rate is the water level rising when the depth f the water is 2 m? 1 2 Recall that, fr a cne, V r h. (Try nt t lk back at yur checklist unless yu are stuck.) r h 5meters
Differentiatin Applicatins 1: Related Rates 157 1. (Check yur wrk) Jeff generates the fllwing slutin t the previus questin: r 2 h 5 (by similar triangles) s h r 25 At the mment f interest: h 2, s Substituting r 0.8 int 1 (0.8) 2 0.67 V h h V 1 2(2) r 0.8 5 2 r h gives dv dh dh dv Differentiating 0.67 and slving gives 1.5 dv Plugging in 1 m dh /hr gives the answer as 1.5 m/hr a. (Check yur wrk) Jeff did a gd jb setting up the variables; and his equatin is crrect. Are these chices cnsistent with yur entries in the table n the previus page? If nt, g back and check yur wrk. r 2h 5 b. The crrect answer t this questin is dh 25 0.5 16 m/hr Jeff made a very cmmn errr fr this type f prblem. He eliminated a variable by substituting in an actual value t sn. Find Jeff s errr, crrect it, and slve the prblem t get the right answer. Shw yur wrk. 2h c. (Check yur wrk) T slve this prblem Jeff shuld have substituted r in the 5 vlume equatin befre using implicit differentiatin (nt the value 0.8). Is this cnsistent with yur answer t part b? If nt, g back and revise yur wrk.
158 Differentiatin Applicatins 1: Related Rates Ntes