The document discusses calculating rates of change for different scenarios involving spherical objects. It provides an example of calculating the rate of change of volume of a deflating balloon when the radius is decreasing at a constant rate. It also works through an example of calculating the rate of increase of the surface area of an expanding bubble when the volume is increasing at a constant rate. The key steps involve relating the rates of change of variables like volume, radius, and surface area using their equations.

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*