Handout 4.11 Solutions. Let x represent the depth of the water at any time Let V represent the volume of the pool at any time

Transcription

1 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 V The level of the water i riing at the rate of m.

2 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 V 7 The edge of the cube i decreaing at the rate of m 7 min.

3 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 A da 6 da da da F bg HG I K J The urface area of the cube i decreaing at the rate of 8 m min.

4 . Label h r Let r repreent the radiu of the cone in Let h repreent the height of the and in Let V repreent the volume of the cone min dh when h=6 V r h r h dh dh Firt write V in term h V r h F H G I h K J h h Then we have dh h 4 48 dh 6 4 dh dh The altitude of the cone i riing at the rate of 4 min.

5 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 y 0 y dy y dy 40 b g Remember when =50 y 50 y y y y The ditance between the train and tudent i decreaing at the rate of 7 km hr.

6 5. Label 4 0 r h Let r repreent the radiu of the top of coffee in Let h repreent the height of the coffee in Let V repreent the volume of the filter 4 dh when h=4 V r h h 0, r 4 dh dh r h 5 Firt write V in term h V r h F H G I h K J 5 4 h 75 h Then we have dh dh dh 4 h 75 dh The altitude of the coffee i falling at the rate of 5 6.

7 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 y=5 m y y dy 0 y 5 dy 5 4 Remember when y=5 y 69 y The top of the ladder i liding down at the rate of 5 4 m.

8 7. Label B C A Let A repreent the area of the triangle in quare unit dy unit da when 7 time econd 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 da dy dy y y y The area i increaing at the rate of 7 unit.

9 8. Label Let r repreent the radiu of the balloon in Let V repreent the volume of the balloon in cubic 00 dr when r=9 V 4 r dr dr 4r dr dr dr : The radiu i increaing at the rate of 5 8.

10 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 y 4 y dy dy y 5 5 Remember when y= y y 0 5 The length of rope i changing at the rate of 5 m.

11 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 y=45 ft y 90 y dy y dy Remember when y=45 y The length of rope i changing at the rate of 0 5 ft.

12 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 y=90 ft y 90 y dy y dy Remember when y=90 y The length of rope i changing at the rate of 50 ft.

13 . a) A and r are related by A r da r dr da dr b) Since D=r then D A D 4 da D dd 6 c) You know that C r, therefore r C Then da dc da dr dr dc r r 6 d) da da dr dr dr r 6 6

14 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 r=0 A r A R r : dr dr da da 8000 r : The rate of rotation i 0 rev/min

15 . 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. : h0 5t : when t= : 0 0 h : 0 0 h h 0h 0 0h 0 h dh dh 00 h 0h 0 h t : The ball hadow i moving along the ground at 0 m

16 4 Solution: Label: Let h repreent the height of the water in m. Let V repreent the volume of the tank in. 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 : A ha b h 0. b V 6 h0. b h 0. 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 h0. 8

17 Thi give u V h 0. h 0. 4 h0.4 h 4 9.h h 4 Now, let find ome derivative dh dh 9 dh. h 9 dh dh : Therefore the height of the water change at m/min