MCB4UW Handout 4.11 Related Rates of Change

Transcription

1 MCB4UW Handout 4. Related Rate of Change. Water flow into a rectangular pool whoe dimenion are m long, 8 m wide, and 0 m deep. If water i entering the pool at the rate of cubic metre per econd (hint: thi i the rate of change in volume), how fat i the level of the water riing? ( hint: let repreent how deep the water i at any pecific time). A chemical cube i left out to dry, the drying proce ymmetrically compact the cube o that the volume decreae at a rate of cubic metre per minute. a) Find the rate of change of an edge of the cube when the volume i 7 cubic metre. b) What i the rate of change of the urface of the cube at thi point?. In the bottom of an hourgla, a conical pile of and i formed at the rate of cubic cm per minute. The radiu of the bae of the pile i alway equal to one-half it altitude. How fat i the altitude riing when it i 6 cm deep? (note: volume of a cone i equal to π rh, where r i the radiu and h i the height of the cone) 4. A math tudent i tanding 0 metre from a traight ection of railroad track. A train i approaching, moving along the track at 90 kilometre per hour. How fat i the ditance between the train and the tudent decreaing when the train i 50 metre from the tudent? 5. A coffee maker ue a filter in the hape of a cone, with the filter being 0 cm high and having a radiu of 4cm. Coffee i flowing from the filter into a cup at a rate of 4 cm per econd. At what rate i the level of coffee in the filter falling when the coffee in the filter i 4 cm deep? 6. One end of a metre ladder i on the ground, and the other end ret on a vertical wall. If the bottom end i drawn away from the wa at metre per econd, how fat i the top of the ladder liding down the wall when the bottom of the ladder i 5 metre from the wall?

2 a 7. Conider a variable right angle triangle ABC in a rectangular coordinate ytem. Verte A i the origin, the right angle i at 7 verte B on the y ai, and verte C i on the parabola y +. If 4 B tart at (0, ) and move upward at a contant rate of unit per 7 econd. How fat i the area of the triangle increaing when t econd? 8. A balloon in the hape of a phere i being inflated o that the volume i increaing by 00 cubic centimetre per econd. At what rate i the radiu increaing when the radiu i 9 cm? 9. From the edge of a dock 4 metre above the urface of the water, a rowboat i being hauled in by a rope and i approaching the bae of the dock at the rate of metre per econd. How fat i the length of rope changing when the boat i metre from the dock? 0. A baeball diamond i a 90 foot quare. A ball i batted along the third-bae line at a contant peed of 00 feet per econd. How fat i it ditance from firt-bae changing when: a) It i halfway to third-bae? b) It reache third-bae?. Let A, D, C, and r be the area, diameter, circumference, and radiu of dr cm a circle, repectively. At a certain intant, r6 and. Find the rate of change of A with repect to: a) r b) D c) C d) t. The head of a hort-ditance radar i et to weep out an area of 8000 km/min. If the beam i et for a ditance of 0km, then find the rate of rotation (in rev/min) of the radar head. A light i at the top of a 0m pole. A ball i dropped from the ame height from a point 0m from the light. The height of the ball (in metre) t econd after it ha been dropped i approimated by h 0 5t. In which direction doe the hadow of the ball move along the ground? How fat i the hadow of the ball moving along the ground later? 4 A water trough i 6m long, and it cro-ection ha the hape of an iocele trapezoid that i 0cm wide at the bottom, 50cm wide at the top and 40cm high. If the trough i being filled with water at the rate of 0. m/min, how fat i the water riing when the water i 5 cm deep. b r h

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4 MCBUW Handout 4. Solution. Label 0m m 8m Let repreent the depth of the water at any time Let V repreent the volume of the pool at any time cm Relationhip V Concluion The level of the water i riing at the rate of m.

5 a). Label metre Let repreent the length of a ide of the cube in metre Let V repreent the volume of the cube at any time m min when Volume i 7 cubic metre, or the ide i metre Relationhip V b g 7 Concluion The edge of the cube i decreaing at the rate of m 7 min.

6 b). Label metre Let repreent the length of a ide of the cube in metre Let V repreent the volume of the cube at any time Let A repreent the urface area of the cube m min m 7 min da when Volume i 7 cubic metre, or the ide i metre Relationhip A da 6 da da da F bg HG I K J Concluion The urface area of the cube i decreaing at the rate of 8 m min.

7 . Label h r Let r repreent the radiu of the cone in cm Let h repreent the height of the and in cm Let V repreent the volume of the cone cm min dh when h6 cm Relationhip V πr r h dh h dh Firt write V in term h V πr h Then we have F H G I π h K J h πh dh dh πh 4 48 dh b g π 6 4 π dh Concluion The altitude of the cone i riing at the rate of 4 cm π min.

8 4. Label 0m y Let repreent the ditance from train to tudent in m Let y repreent the horizontal track ditance in m dy m 90, 000 hr when 50 m Relationhip y +0 y dy y dy 40 b g Remember when 50 y + 0 b g 50 y y y 600 y 40 Concluion The ditance between the train and tudent i decreaing at the rate of 7 km hr.

9 5. Label 4 0 r h Let r repreent the radiu of the top of coffee in cm Let h repreent the height of the coffee in cm Let V repreent the volume of the filter cm 4 dh when h4 cm Relationhip V πr h h 0, r h r 4 5 dh dh Firt write V in term h V πr h Then we have F H G I π h K J h 5 4 πh 75 dh dh dh 4 πh 75 dh 4 75 π 4 b gb g 5 6π Concluion The altitude of the coffee i falling at the rate of 5 cm 6π.

10 6. Label m y Let repreent the height of the ladder on the wall in m Let y repreent the ditance the bottom of the ladder from the wall in m dy m when y5 m Relationhip + y + y dy 0 y 5 dy b g 5 4 Remember when y5 + y 69 y Concluion The top of the ladder i liding down at the rate of 5 4 m.

11 7. Label B C A Let A repreent the area of the triangle in quare unit dy unit da 7 when time econd Relationhip 7 y ( y ) 4 A y y 7 ( ) When t then y ( ) Since B tart at y and move up at unit/econd for 7 econd Concluion da dy dy ( y ) y ( y ) ( ) ( 8 ) ( 8) ( 8 ) ( ) The area i increaing at the rate of 7 unit.

12 8. Label Let r repreent the radiu of the balloon in cm Let V repreent the volume of the balloon in cubic cm cm 00 dr when r9 cm Relationhip V 4 π r dr dr dr 4πr dr 00 4π 9 5 8π b g dr Concluion The radiu i increaing at the rate of 5 cm 8.

13 9. Label y 4m Let repreent the ditance from the dock in m Let y repreent the length of rope in m m dy when m Relationhip y + 4 y dy dy y b g 5 5 Remember when y y + 4 bg y 0 5 Concluion The length of rope i changing at the rate of 5 m.

14 0a. Label y Let repreent the ditance from firt bae ft Let y repreent the ditance from home plate in ft dy 00 ft when y45 ft Relationhip y +90 y dy y dy b g Remember when y45 y + 90 b g Concluion The length of rope i changing at the rate of 0 5 ft.

15 0b. Label y Let repreent the ditance from firt bae ft Let y repreent the ditance from home plate in ft dy 00 ft when y90 ft Relationhip y +90 y dy y dy b g Remember when y90 y + 90 b g Concluion The length of rope i changing at the rate of 50. ft

16 . a) A and r are related by A π r da π r dr When r6 then b) Since Dr then da cm π. dr D A π π D 4 da π D dd ( ) π cm 6π c) You know that C π r, therefore r C π Then da dc da dr i dr dc π r π r 6 cm d) da da dr dr dr π r π 6 6 ( )( ) cm

17 Label: Let A repreent the area Let r repreent the radiu Let R repreent the number of revolution Let t repreent the time in minute da : 8000 : dr when r0 Relationhip: A π r R π A ( r) : dr dr da da 8000 π r π 0 π Concluion: The rate of rotation i 0 π rev/min

18 . Solution: 0m h 0m From the contet of the quetion the hadow i moving toward the pole. Label: Let repreent the length of hadow Let h repreent the height of the ball Let t repreent the time in econd. : h 0 5t : when t Relationhip: h : h h + 0h 0 0h 0 h dh dh [ 0]( 0 h) ( 0h)[ ] [ 0t ] ( 0 h) ( 0)( 5) + ( 0)( 5) ( 0 ) 5 0 Concluion: The ball hadow i moving along the ground at 0 m

19 4 Solution: Label: Let h repreent the height of the water in m. Let V repreent the volume of the tank in cm. Let r repreent the wih of the triangle in m. Let a repreent the hort length of the trapezoid in m. Let b repreent the long length of the trapezoid in m. Let t repreent the time in minute : m 0. min : dh when h 0.5 m h + Relationhip: A h( a+ b) ( 0. b) V 6 h 0. + h ( b) ( b) r h : We need to get V in term of one variable h. 0.5 r 0.4 h 0.5h 0.4r r h 8 b h

20 Thi give u V h 0. + h h h 4 9.h+ h 4 Now, let find ome derivative dh dh 9 dh. + h 9 dh ( 0.5) dh Concluion: Therefore the height of the water change at m/min