Related Rates
Finding Related Rates
● Problem Solving with
Related Rates
1
Related Rates – Solving Differential
Equations
Find the indicated values for dy/dt and dx/dt.
Given : y  x  2 x
2


dy
dx
Find
when x  3 and
2
dt
dt
d
 y   d x 2  2 x  dy  2 x  2 dx
dt
dt
dt
dt
dx
dy
When x  3 and
 2, you have
 23  22  8
dt
dt
2
Guidelines for Solving Related-Rate
Problems




3
Identify all given quantities and quantities to be
determined. Make a sketch and label the quantities.
Write an equation involving the variables whose
rates of change either are given or are to be
determined
Using the Chain Rule, implicitly differentiate both
sides of the equation with respect to time t.
After completing Step 3, substitute into the resulting
equation all known values for the variables and their
rates of change. Then solve for the required rate of
change.
Filling a Spherical Balloon
A spherical balloon is inflated with gas at the
rate of 20 ft3/min. Find how fast is the radius of
the balloon increasing at the instant the radius
is
a) 1 ft
b) 2 ft
4
Organize, Identify, and Write an Equation
4 3
dr
dv
V  r , find
when
 20, and r  1, 2
3
dt
dt
d
d 4 3
dV
2 dr
V    r    4r
Differenti ate
dt
dt  3
dt
dt

dV
2 dr
 20  4 (1)
Substitute 20 for
, 1 for r
dt
dt
dr 20 5
dr



Solve for
, Simplify
dt 4 
dt
5
The radius is increasing at ft/min whe n the radius is 1 ft.

5
When 2 is substitute d for r , the radius is incrasing at
ft/min
2
5
Filling a Conical Tank
A water tank has the shape of an
inverted cone with base radius of
2 m and height of 4 m. If water is
being pumped into the tank at a
rate of 2 m3/min, find the rate at
which the water level is rising
when the water is 3 m deep.
6
2m
4m
3m
Organize, Identify, and Write an Equation
1
dV
dh
V  Bh ,
 2, find
when h  3.
3
dt
dt
Since 2r = h, and r = h/2, substitute h/2 for r in order to
have an equation 2in just V and h.
1 2
1 h
1 3
V  r h     h  h
3
3 2
12
7
Volume equation
d
d  1 3
dV 1 2 dh
V    h    h
Differenti ate
dt
dt 12
dt 4
dt

1
9 dh
dh
2 dh
2   3
2
Substitute , solve for
4
dt
4 dt
dt
dh 4
8
8

2 
The water level is rising at a rate of
meters/min
dt 9
9
9
when the height is 3 m
Riding a Bike
At a certain moment, one bicyclist is 4 miles east of an
intersection, traveling towards the intersection at the
rate of 9 mi/hr. At the same time, a second bicyclist is
3 miles south of the intersection traveling away from
the intersection at a rate of 10 mi/hr. At that moment,
at what rate is the distance between the two cyclist
9 mi/hr
increasing or decreasing?
4 mi
3 mi
10 mi/hr
8
y x
z 5 mi
Organize, Identify, and Write an Equation
dx
dy
dz
 9,
 10. Find
dt
dt
dt
when x  4 and y  3. Use the Pythagorea n Theorem.
From the informatio n given let
 
 
 
d 2
d 2
d 2
dx
dy
dz
x 
y 
z  2x  2 y
 2z
Differenti ate Implicitel y
dt
dt
dt
dt
dt
dt
dz
72   60  dz
dz
24 9   23 10   25 

Substitute , solve for
dt
10
dt
dt
dz 12

 1.2
The distance between th e bicyclists is increasing at a rate of
dt 10
1.2 miles/hour at that moment.
9