1. Rates of Change

2. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.

3. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt

4. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.

5. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV
?
dt

6. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV
?
dt
dr
 10
dt

7. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV dV dr dV
?  
dt dt dt dr
dr
 10
dt

8. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV 4 3 dV dr dV
? V  r  
dt 3 dt dt dr
dr dV
 10  4r 2
dt dr

9. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV 4 3 dV dr dV
? V  r  
dt 3 dt dt dr
dr
 10
dV
 4r 2  104r 2 
dt dr
 40r 2

10. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV 4 3 dV dr dV when r  100, dV  40 100 2
? V  r   dt
dt 3 dt dt dr
 400000
dr
 10
dV
 4r 2  104r 2
dt dr
 40r 2

11. Rates of Change
In some cases two, or more, rates must be found to get the
equation in terms of the given variable.
dy dy dx
 
dt dx dt
e.g. (i) A spherical balloon is being deflated so that the radius
decreases at a constant rate of 10 mm/s.
Calculate the rate of change of volume when the radius of the
balloon is 100 mm.
dV 4 3 dV dr dV when r  100, dV  40 100 2
? V  r   dt
dt 3 dt dt dr
 400000
dr
 10
dV
 4r 2  104r 2
dt dr
 40r 2
 when the radius is 100 mm, the volume is decreasing at a
rate of 400000mm 3 / s

12. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?

13. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS
?
dt

14. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV
?  70
dt dt

15. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV
?  70
dt dt
dS dV dS dr
  
dt dt dr dV

16. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV 4 3
?  70 V  r
dt dt 3
dV
 4r 2
dr
dS dV dS dr
  
dt dt dr dV

17. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV 4 3
?  70 V  r S  4r 2
dt dt 3
dS
dV  8r
 4r 2 dr
dr
dS dV dS dr
  
dt dt dr dV

18. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV 4 3
?  70 V  r S  4r 2
dt dt 3
dS
dV  8r
 4r 2 dr
dr
dS dV dS dr
  
dt dt dr dV
 1 
  70  8 r   
 4 r 2 
140

r

19. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV 4 3
?  70 V  r S  4r 2
dt dt 3
dS
dV  8r
 4r 2 dr
dr
dS dV dS dr dV 140
   when r  10, 
dt dt dr dV dt 10
 1   14
  70  8 r   
 4 r 2 
140

r

20. (ii) A spherical bubble is expanding so that its volume increases at a
constant rate of 70 mm 3 /s
What is the rate of increase of its surface area when the radius is
10 mm?
dS dV 4 3
?  70 V  r S  4r 2
dt dt 3
dS
dV  8r
 4r 2 dr
dr
dS dV dS dr dV 140
   when r  10, 
dt dt dr dV dt 10
 1   14
  70  8 r   
 4 r 2   when radius is 10mm the surface area is
140 increasing at a rate of 14mm 2 / s

r

21. Exercise 7E; 2, 5, 6, 9, 13*
Exercise 7F; 2, 5, 9, 10, 11*