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
x
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?

3. Rates of Change
x ?
dV
dt

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?

4. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
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?

5. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
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?

6. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

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?

7. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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?

8. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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 1
3
10
x 

9. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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 1
3
10
x 
2
3
10
x
 

10. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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 1
3
10
x 
2
3
10
x
 
 
2
3 5
when 5,
10
dV
x
dt
  

11. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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 1
3
10
x 
2
3
10
x
 
 
2
3 5
when 5,
10
dV
x
dt
  
7.5 

12. Rates of Change
x ?
dV
dt

1
10
dx
dt
 
3
V x
2
3
dV
x
dx

dV dV dx
dt dx dt
 
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 1
3
10
x 
2
3
10
x
 
 
2
3 5
when 5,
10
dV
x
dt
  
7.5 
3
volume is decreasing at 7.5 cm /s

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

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

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

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

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

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

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

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

21. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x

22. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /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
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /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
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt


25. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x

26. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx


27. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 

28. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 

29. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 
2
1
5 x


30. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 
2
1
5 x

 
2
1
when 4,
5 4
dx
x
dt 
 

31. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 
2
1
5 x

 
2
1
when 4,
5 4
dx
x
dt 
 
1
80


32. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 
2
1
5 x

 
2
1
when 4,
5 4
dx
x
dt 
 
1
80

depth is increasing
1
at cm/min
80


33. (ii) A vessel is in the form of an inverted cone with a vertical angle of
If the depth of the water in the vessel is x cm;
a) find the volume of water.
90
x
x
45
x 21
3
V r h
21
3
x x
31
3
x
3
b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?
dx
dt

1
5
dV
dt

31
3
V x
2dV
x
dx

dx dx dV
dt dV dt
 
2
1 1
5x
 
2
1
5 x

 
2
1
when 4,
5 4
dx
x
dt 
 
1
80

depth is increasing
1
at cm/min
80

Exercise 7H;
1a, 2a, 4,
6, 7, 8, 10,
12, 14