2. Ex. Two rates that are related.
Related rate problems are differentiated with
respect to time. So, every variable, except t is
differentiated implicitly.
Given y = x2
+ 3, find dy/dt when x = 1, given
that dx/dt = 2.
y = x2
+ 3
dt
dx
x
dt
dy
2=
Now, when x = 1 and dx/dt = 2, we
have
4)2)(1(2 ==
dt
dy
3. Procedure For Solving
Related Rate Problems
1. Assign symbols to all given quantities and
quantities to be determined. Make a sketch
and label the quantities if feasible.
2. Write an equation involving the variables
whose rates of change either are given or are
to be determined.
3. Using the Chain Rule, implicitly differentiate
both sides of the equation with respect to t.
4. Substitute into the resulting equation all known
values for the variables and their rates of change.
Solve for the required rate of change.
4. Ex. A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The radius
r of the outer ripple is increasing at a constant rate
of 1 foot per second. When this radius is 4 ft., what
rate is the total area A of the disturbed water
increasing.
Given equation: 2
rA π=
Givens: 41 == rwhen
dt
dr
Differentiate:
dt
dr
r
dt
dA
π2= ( )( )412π=
dt
dA
π8=
?=
dt
dA
5. An inflating balloon
Air is being pumped into a spherical balloon at the
rate of 4.5 in3
per second. Find the rate of change
of the radius when the radius is 2 inches.
Given:
sec/5.4 3
in
dt
dV
= r = 2 in. ?: =
dt
dr
Find
Equation: 3
3
4
rV π=
Diff.
& Solve: dt
dr
r
dt
dV 2
4π=
dt
dr2
245.4 π=
.09in/sec=
dr
dt
6. The velocity of an airplane tracked by radar
An airplane is flying at an elevation of 6 miles on a flight
path that will take it directly over a radar tracking station.
Let s represent the distance (in miles)between the radar
station and the plane. If s is decreasing at a rate of 400
miles per hour when s is 10 miles, what is the speed of
the plane.
7. Given:
Find:
Equation:
Solve:
10400 =−= s
dt
ds
?=
dt
dx
x2
+ 62
= s2
dt
ds
s
dt
dx
x 22 =
To find dx/dt, we
must first find x
when s = 10
836100362
=−=−= sx
( ) ( )( )40010282 −=
dt
dx mph
dt
dx
500−=
Speed is absolute value of velocity = 500 mph
8. A fish is reeled in at a rate of 1 foot per second
from a bridge 15 ft. above the water. At what
rate is the angle between the line and the water
changing when there is 25 ft. of line out?
15 ft.
x
θ
9. Given:
Find:
Equation:
Solve:
1−=
dt
dx
x = 25 ft. h = 15 ft.
?=
dt
dθ
x
15
sin =θ
1
15sin −
= xθ
( )
dt
dx
x
dt
d 2
15cos −
−=
θ
θ
dt
dx
xdt
d
θ
θ
cos
15
2
−
=
( )1
25
20
25
15
2
−
−
=
dt
dθ
sec/
100
3
rad
dt
d
=
θ
10. Ex. A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The radius
r of the outer ripple in increasing at a constant rate
of 1 foot per second. When this radius is 4 ft., what
rate is the total area A of the disturbed water
increasing.
An inflating balloon
Air is being pumped into a spherical balloon at the
rate of 4.5 in3
per minute. Find the rate of change
of the radius when the radius is 2 inches.
