1. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?

2. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
x

3. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
x ?
dt

4. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV dx 1
x ? 
dt dt 10

5. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10

6. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx

7. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
dV dV dx
 
dt dx dt

8. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
dV dV dx
 
dt dx dt
1
 3x  
2
10

9. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
dV dV dx
 
dt dx dt
1
 3x  
2
10
3x 2

10

10. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
3 5
2
dV dV dx dV
  when x  5, 
dt dx dt dt 10
1
 3x  
2
10
3x 2

10

11. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
3 5
2
dV dV dx dV
  when x  5, 
dt dx dt dt 10
 3x  
2 1  7.5
10
3x 2

10

12. Rates of Change
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
dV
?
dx

1 V  x3
x
dt dt 10 dV
 3x 2
dx
3 5
2
dV dV dx dV
  when x  5, 
dt dx dt dt 10
 3x  
2 1  7.5
10
3x 2  volume is decreasing at 7.5 cm3 /s

10

13. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.

14. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.

15. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x

16. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x

17. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x
45
x

18. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x
x
45
x

19. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45
x

20. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3

21. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3

22. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.

23. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx
?
dt

24. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx
?
dt
dV 1

dt 5

25. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx
?
dt
dV 1

dt 5
1
V   x3
3

26. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx
?
dt
dV 1

dt 5
1
V   x3
dV 3 2
x
dx

27. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV
?  
dt dt dV dt
dV 1

dt 5
1
V   x3
dV 3 2
x
dx

28. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV
?  
dt dt dV dt
dV 1
 1 1
dt 5  2
1 3 x 5
V  x
dV 3 2
x
dx

29. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV
?  
dt dt dV dt
dV 1
 1 1
dt 5  2
1 3 x 5
V  x 1
dV 3 
x 2
5 x 2
dx

30. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV dx 1
?   when x  4, 
dt dt dV dt dt 5  4 2
dV 1
 1 1
dt 5  2
1 3 x 5
V  x 1
dV 3 
x 2
5 x 2
dx

31. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV dx 1
?   when x  4, 
dt dt dV dt dt 5  4 2
dV 1
 1 1 
1
 2
dt 5
x 5 80
1
V   x3 1
dV 3 
x 2
5 x 2
dx

32. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV dx 1
?   when x  4, 
dt dt dV dt dt 5  4 2
dV 1
 1 1 
1
 2
dt 5
x 5 80
1
V   x3  depth is increasing
1
dV 3  1
x 2
5 x 2 at cm/min
dx 80

33. (ii) A vessel is in the form of an inverted cone with a vertical angle of 90
If the depth of the water in the vessel is x cm;
a) find the volume of water.
x 1 2
V  r h
x  3
45 1 2
 x x
x 3
1 3
 x
3
b) If water is poured in at a rate of 0.2 cm3 /min, find the rate the depth
is increasing when the water depth is 4 cm.
dx dx dx dV dx 1
?   when x  4, 
dt dt dV dt dt 5  4 2 Exercise 7H;
dV 1 1a, 2a, 4,
 1 1 
1
 2
dt 5
x 5 80 6, 7, 8, 10,
1
V   x3  depth is increasing 12, 14
1
dV 3 2  1
x 5 x 2 at cm/min
dx 80