12 x1 t04 01 rates of change (2012)
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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*
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