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2.6 Related Rates I
Volume changes-implicit• With change in volume, volume, radius, and height of the
volume in a cone all depend upon the time of the change occurring.
• Volume of the cone depends on r and h• Differentiate with respect to t (related rate equation)
23V r h
23
2
2
( )
(2 )3
23
d dV r h
dt dtdV dh dr
r h rdt dt dt
dV dh drr rh
dt dt dt
X and y with respect to t• Given find when x=3 with
and find when x=1 with
22( 3 )y x x dy
dt2
dx
dt
dx
dt5
dy
dt
Answer• Differentiate with respect to t
2[ ] [2( 3 )]
2[2 3 ]
2[2(3)(2) 3(2)] 12
d dy x x
dt dtdy dx dx
xdt dt dtdy
dt
2[ ] [2( 3 )]
2[2 3 ]
2 [2 3] [4 6]
[4 6]
5 5
[4(1) 6] 2
d dy x x
dt dtdy dx dx
xdt dt dtdy dx dx
x xdt dt dtdy
dxdtx dt
dx
dt
Rate of change• The radius of a circle increases at a rate of 3cm/min. Find the
rate of change of the area when r=6cm.
Answer• Differentiate with respect to t
2
2
2
36min
3
[ ] [ ]
(2 )
(2(6)(3)) cm
drA r
dtd dA r
dt dtdA dr
rdt dtdA
dt
Rate of increase• A spherical balloon is inflated at the rate of 800 cubic cm/min.
How fast is the radius increasing at the instant the radius is 30 cm?
Answer• Differentiate with respect to time
343
343
243
2
2
min2
800
[ ]
(3 )
4 ( )
800 4 ((30) )
800 2
4 (30) 9cm
dVV r
dtd dV r
dt dtdV dr
rdt dtdV dr
rdt dt
dr
dtdr
dt
