Rates of Change

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

Rates of Change

dtdx

dxdy

dtdy

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

Rates of Change

dtdx

dxdy

dtdy

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

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.

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

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.

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

drdV

dtdr

dtdV

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

2

3

4

34

rdrdV

rV

drdV

dtdr

dtdV

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

2

3

4

34

rdrdV

rV

drdV

dtdr

dtdV

2

2

40410r

r

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

2

3

4

34

rdrdV

rV

drdV

dtdr

dtdV

2

2

40410r

r

400000

10040,100when 2

dtdVr

Rates of Change

dtdx

dxdy

dtdy

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

?dtdV

10dtdr

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.

2

3

4

34

rdrdV

rV

drdV

dtdr

dtdV

2

2

40410r

r

400000

10040,100when 2

dtdVr

s/mm400000 of rate aat decreasingisvolumethemm,001isradiuswhen the

3

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

/smm70 3

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

/smm70 3

?dtdS

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

/smm70 3

?dtdS 70

dtdV

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

/smm70 3

?dtdS 70

dtdV

dS dV dS drdt dt dr dV

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

/smm70 3

?dtdS 70

dtdV

2

3

4

34

rdrdV

rV

dS dV dS drdt dt dr dV

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

/smm70 3

?dtdS 70

dtdV

2

3

4

34

rdrdV

rV

rdrdS

rS

8

4 2

dS dV dS drdt dt dr dV

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

/smm70 3

?dtdS 70

dtdV

2

3

4

34

rdrdV

rV

rdrdS

rS

8

4 2

dS dV dS drdt dt dr dV

2

170 84

140

rr

r

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

/smm70 3

?dtdS 70

dtdV

2

3

4

34

rdrdV

rV

rdrdS

rS

8

4 2

14 10

140,10when

dtdVr

dS dV dS drdt dt dr dV

2

170 84

140

rr

r

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

/smm70 3

?dtdS 70

dtdV

2

3

4

34

rdrdV

rV

rdrdS

rS

8

4 2

14 10

140,10when

dtdVr

s/14mm of rate aat increasing isareasurfacethe10mmisradiuswhen

2

dS dV dS drdt dt dr dV

2

170 84

140

rr

r

Exercise 7E; 2, 5, 6, 9, 13*

Exercise 7F; 2, 5, 9, 10, 11*