4.6 Related Rates 8 b o lk0uct5ai FSHopfitcwkadr9ee MLBL1Cv.0 h ca5lrlx 8rzi8gThzt Zs9 2rJejsqeprTvCeVdy.w Solve each related rate problem.
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1 -- g 52P0l33e 5Ktu3tlaY tswobfrtcwsawrkeq mlzlzcd.u 2 7AklGlf lrbiegkhjtbsa 9rlewsSeIr2vPeVdW.L 2 7Mza5dWeI gwbimtmhn bimnff0ieneistuet SCDallJcrulsuTsG.k Calculus 4.6 Related Rates 8 b o lk0uct5ai FSHopfitcwka9ee MLBLCv.0 h ca5lrlx 8rzi8gThzt Zs9 2rJejsqeprTvCeVdy.w Solve each related rate problem. Name Date ) A spherical balloon is deflated so that its radius decreases at a rate of 4 cm/sec. At what rate is the volume of the balloon changing when the radius is 3 cm? Period 2) A spherical balloon is deflated at a rate of 256π 3 balloon changing when the radius is 8 cm? cm³/sec. At what rate is the radius of the 3) Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 9 cm/min. How fast is the area of the pool increasing when the radius is 2 cm? 4) A 7 ft tall person is walking towards a 7 ft tall lamppost at a rate of 4 ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 2 ft from the lamppost? 5) A conical paper cup is 30 cm tall with a radius of 0 cm. The cup is being filled with water at a rate of 2π 3 cm³/sec. How fast is the water level rising when the water level is 2 cm? 6) A 3 ft ladder is leaning against a wall and sliding towards the floor. The top of the ladder is sliding down the wall at a rate of 7 ft/sec. How fast is the base of the ladder sliding away from the wall when the base of the ladder is 2 ft from the wall? 7) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 2 m/min. How fast is the area of the spill increasing when the radius is 3 m? 8) A hypothetical cube shrinks so that the length of its sides are decreasing at a rate of 2 m/min. At what rate is the volume of the cube changing when the sides are 2 m each? 9) A conical paper cup is 0 cm tall with a radius of 0 cm. The bottom of the cup is punctured so that the water level goes down at a rate of 2 cm/sec. At what rate is the volume of water in the cup changing when the water level is 9 cm? 0) An observer stands 500 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 700 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 200 ft from the ground? ) A spherical snowball melts at a rate of 36π in³/sec. At what rate is the radius of the snowball changing when the radius is 5 in?
2 -2- d w2e0wn3n xkpu8t5ae JSuo2fqtowza9rSeK pl4licv.p O 2A9lZl6 ir6idghhotds7 br0eesweprvvoeydf. O WMaaidGeH PwTimtMh7 ZI4nWfniNnciOtIeI MCva7lJcJuuleuMs9.A 2) A hypothetical cube grows at a rate of 8 m³/min. How fast are the sides of the cube increasing when the sides are 2 m each? 3) A conical paper cup is 0 cm tall with a radius of 30 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 9 cm? 4) Water slowly evaporates from a circular shaped puddle. The radius of the puddle decreases at a rate of 8 in/hr. Assuming the puddle retains its circular shape, at what rate is the area of the puddle changing when the radius is 3 in? 5) A hypothetical square grows so that the length of its diagonals are increasing at a rate of 4 m/min. How fast is the area of the square increasing when the diagonals are 4 m each? 6) Water slowly evaporates from a circular shaped puddle. The area of the puddle decreases at a rate of 6π in²/hr. Assuming the puddle retains its circular shape, at what rate is the radius of the puddle changing when the radius is 2 in? 7) A hypothetical cube grows so that the length of its sides are increasing at a rate of 4 m/min. How fast is the volume of the cube increasing when the sides are 7 m each? 8) A hypothetical square grows at a rate of 6 m²/min. How fast are the sides of the square increasing when the sides are 5 m each? 9) A hypothetical cube shrinks at a rate of 8 m³/min. At what rate are the sides of the cube changing when the sides are 3 m each? 20) A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 8 in? 2) A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of 2 mm/sec. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 2 mm each? 22) A conical paper cup is 0 cm tall with a radius of 0 cm. The bottom of the cup is punctured so that the water leaks out at a rate of 9π 4 cm³/sec. At what rate is the water level changing when the water level is 6 cm? 23) A hypothetical square shrinks so that the length of its diagonals are changing at a rate of 8 m/min. At what rate is the area of the square changing when the diagonals are 5 m each? 24) A hypothetical square shrinks at a rate of 2 m²/min. At what rate are the diagonals of the square changing when the diagonals are 7 m each? 25) Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 25π cm²/min. How fast is the radius of the pool increasing when the radius is 6 cm?
3 e b2o0ks33 RK6urttaG xsvofytqwwaeree8 dlalicp.j P 0Aolln jrgibgjhhtesy LrCeEsoeAvuejdz.m h smqaodwez TwWixtvhq RI lnkfnigngirtwef ac3afldcjuclauysc.k -3- Answers to 4.6 Related Rates ) V volume of sphere r radius t time Equation: V 4 3 πr3 Given rate: 4 r 3 4πr 2 44π cm³/sec 2) V volume of sphere r radius t time Equation: V 4 3 πr3 Given rate: 256π 3 r 8 4πr 2 3 cm/sec 3) A area of circle r radius t time Equation: A πr 2 Given rate: 9 r 2 2πr 26π cm²/min r 3 r 2 4) x distance from person to lamppost y length of shadow t time Equation: x + y 7 y Given rate: dx 7 4 dy dy x dx 4 5 ft/sec 5) V volume of material in cone h height t time Equation: V πh3 Given rate: 2π dh 27 3 dh h 2 9 πh cm/sec 6) x horizontal distance from base of ladder to wall y vertical distance from top of ladder to floor t time Equation: x 2 + y Given rate: dy 7 dx dx x 2 y x dy 9 2 ft/sec 7) A area of circle r radius t time Equation: A πr 2 Given rate: 2 r 3 2πr 52π m²/min r 3 x 2 h 2 r 8 x 2
4 j C280aC33 xksuktyaq XS4oef9tGw2aPrVeY 4LJLqCc.N h 0AplmlQ rcizgphttlsg PrneOserrWv7eMdJ.8 p MMOaTdheh LwpiKtjhn 0IwnNfmijnRiWtce4 0C3a9lSciuK luhs4.x -4-8) V volume of cube s length of sides t time Equation: V s 3 Given rate: ds 2 s 2 3s 2 ds 24 m³/min 9) V volume of material in cone h height t time Equation: V πh3 Given rate: dh 3 2 h 9 πh 2 dh 62π cm³/sec 0) a altitute of rocket z distance from observer to rocket t time Equation: a z 2 Given rate: da 700 dz dz a 200 a z da ft/sec s 2 h 9 ) V volume of sphere r radius t time Equation: V 4 3 πr3 Given rate: 36π r 5 4πr in/s 2) V volume of cube s length of sides t time Equation: V s 3 Given rate: 8 ds ds s 2 3s m/min s 2 3) V volume of material in cone h height t time Equation: V 3πh 3 Given rate: dh 2 h 9 9πh 2 dh 458π cm³/sec 4) A area of circle r radius t time Equation: A πr 2 Given rate: 8 r 3 2πr 48π in²/hr h 9 r 3 r 5 a 200
5 A f2e0jo3f mkeu6teah asto7f9tqwrajrceb hlylcc4.w 8 TAGlylW mrki9gwhetssn zrge0swelr6vpebd.q 8 JM5apdeeo Twoi5t6hL PIinifgiAngiWteg kcpahlkc9u3lquqss.p -5-5) A area of square x length of diagonals t time Equation: A x2 2 x 4 x dx Given rate: dx 4 56 m²/min 6) A area of circle r radius t time Equation: A πr 2 Given rate: 6π r 2 2πr 2 3 in/hr x 4 7) V volume of cube s length of sides t time Equation: V s 3 Given rate: ds 4 s 7 3s 2 ds 588 m³/min 8) A area of square s length of sides t time Equation: A s 2 Given rate: 6 ds ds s 5 2s 8 5 m/min 9) V volume of cube s length of sides t time Equation: V s 3 Given rate: 8 ds ds s 3 3s m/min s 7 20) V volume of sphere r radius t time Equation: V 4 3 πr3 Given rate: 4 r 8 4πr 2 024π in³/sec 2) V volume of cube s length of sides t time Equation: V s 3 Given rate: ds 2 s 2 3s 2 ds 24 mm³/sec s 5 s 3 s 2 r 2 r 8
6 c H2L0ze3 ik9uktbar dsyozfjtlwxaprvec plrldce.q h GANlRlS ArHi0gPhCt7sE Jr ee4sezr6vle2dn.m X MHaves bwpiztwhe 2IynUfPienwi6tQeY ecraalcc2usl3uuso.n -6-22) V volume of material in cone h height t time Equation: V πh3 Given rate: 9π dh 3 4 dh h 6 πh 2 6 cm/sec 23) A area of square x length of diagonals t time Equation: A x2 x 5 2 x dx Given rate: dx 8 40 m²/min 24) A area of square x length of diagonals t time Equation: A x2 2 dx x 7 Given rate: 2 x 2 7 m/min 25) A area of circle r radius t time Equation: A πr 2 Given rate: 25π r 6 2πr 25 2 cm/min dx x 5 x 7 r 6 h 6
Chapter 3 Practice Test
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