Unit 2: Differentiation
Section 2.6: Relate Rates
• Find a Relate Rate.
•Use Related Rates to Solve Real-Life Problems.
Mr. G
Example 1: Two Rates that are Related
Suppose that x and y are both differentiable functions of t and related
by the equation y  x 2  2 x . Find dy/dt when x = 1, given that
dx/dt = 2 when x = 1.
y  x2  2 x
d 2
d
 y    x  2 x 
dt
dt
dx
dx
dy
 2x  2
dt
dt
dt
dy
 2 1 2   2  2   8
dt
Mr. G
Example 2: Volume
The radius r of a sphere is increasing at a rate of 2 inches per second.
1. Find the rate of change of the volume when r = 6 inches and
when r = 24 inches.
4 3
V  r
3
dV
2
 4  6in   2in / sec 
dt
d 4 3
dV
r 


dt
dt  3

dV
 288 in3 sec
dt
dV
2 dr
 4 r
dt
dt
dV
3
 4608 in sec
dt
Mr. G
Example 2: Volume
The radius r of a sphere is increasing at a rate of 2 inches per second.
1. Explain why the rate of change of the volume of the sphere is not
constant even though dr/dt is constant.
dV
2 dr
 4 r
dt
dt
If the dr/dt is constant, dV/dt is proportional to r²
Mr. G
Example 3: Moving Ladder
A ladder 10 ft long rest against a vertical wall. If the bottom of
the ladder slides away from the wall at a rate of 1 ft/s, how fast is
the top of the ladder sliding down the wall when the bottom of
the ladder is 6 ft. from the wall?
dx
Given
 1 ft / sec
dt
Find
dy
?
dt
x 2  y 2  100
2x
dx
dy
 2y
0
dt
dt
dy
x dx

dt
y dt
dy

dt
dx

dt
When x = 6 the Pythagorean
theorem gives y = 8
3
dy
6
  1   ft / sec
4
dt
8
Mr. G
Example 4: Moving Object
A man walks along a straight path at a speed of 4 ft/s. A
searchlight is located on the ground 20 ft from the path and is
kept focused on the man . At what rate is the searchlight rotating
when the man is 15 ft from the point on a path closest to the
searchlight?
d
dx
Find
when x  15 ft
 4 ft / sec
dt
dt
x
tan  
 x  20 tan 
20
dx
d
1
dx
2
2
 20sec 
 cos 
dt
dt
20
dt
1
 cos 2   4 
20
When x  15, the length of the beam is 25, so cos  
4
5
2
d
1

cos 2   4   1  4   4   16  0.128rad / sec
 
dt 20
125
20  5 
Mr. G
Car A is traveling west at 50 mi/h and car B is traveling north at 60
mi/h. Both are headed for the intersection of the two roads. At what
rate are the cars approaching each other when car A is 0.3 mi and car
B is 0.4 mi from the intersection? C is the intersection of the roads.
At a given time T, let X be the distance from car A to C, let Y be the
distance from car B to C, and let Z be the distance between the cars,
where X,Y, and Z are measured in miles.
dx
dy
mi
 50
and
 60 mi
h
h
dt
dt
Differentiating each side
with respect to t, we have
dz
?
dt
x2  y 2  z 2
dx
dy
1  dx
dy 
dz
dx
dy 2 x dt  2 y dt
 x  y 
2z  2x  2 y

dt
dt
dt
2z
z  dt
dt 
Mr. G
When x  0.3mi and y  0.4mi the pythagoream theorem gives
z  0.5mi
dz 1  dx
dy 
 x  y 
dt z  dt
dt 
dz
1

0.3  50   0.4  60  

dt 0.5
dz
 78mi / h
dt
The cars are approaching each
other at a rate of 78 mi/h
Mr. G
A water tank has the shape of an inverted circular cone with base
radius 2 m and height 4 m. If water is being pumped into the tank at
a rate of 2 cubic meters per minute, find the rate at which the water
level is rising when the water is 3 m deep.
dV
m3
2
dt
min
dh
 ? when h  3 m
dt
The quantities V and h are related by the equation
1
V   r 2h
3
r 2
Using similar triangles we have 
h 4
h
 r
2
1 h
 3
So the new volume expression becomes V     h 
h
3 2
12
2
Mr. G
Now we can differentiate each side with respect to t to:
dV  2 dh
 h
dt
4
dt
dh
4 dV
 2
dt  h dt
3
dV
m
Substituing h  3 m and
2
, we have
min
dt
dh
4

2
2
dt   3
dh 8

dt 9
dh
 0.283 m
min
dt
Mr. G
Example : AP exam Type Question
Ship A is traveling due west toward Lighthouse Rock at a rate of 15
km per hour. Ship B is traveling due north away from Lighthouse at a
speed of 10 km/hr. Let x be the distance between ship A and
Lighthouse Rock at time t, and let y be the distance between ship B
and Lighthouse Rock at time t, as shown in the figure below.
Find the distance, in kilometers,
between ship A and ship B when
x = 4 km and y = 3 km.
a) Distance  32  42  5 km
Mr. G
Example : AP exam Type Question
Ship A is traveling due west toward Lighthouse Rock at a rate of 15
km per hour. Ship B is traveling due north away from Lighthouse at a
speed of 10 km/hr. Let x be the distance between ship A and
Lighthouse Rock at time t, and let y be the distance between ship B
and Lighthouse Rock at time t, as shown in the figure below.
Find the rate of change, in km/hr, of
the distance between the two ships
when x = 4 km and y = 3 km.
r x y
2
2
2
r  x2  y 2
At x = 3 and y = 4
dx
dy
x y
dr
dt
 dt
dt
x2  y 2
dr
 6 km
h
dt
Mr. G
Example : AP exam Type Question
Ship A is traveling due west toward Lighthouse Rock at a rate of 15
km per hour. Ship B is traveling due north away from Lighthouse at a
speed of 10 km/hr. Let x be the distance between ship A and
Lighthouse Rock at time t, and let y be the distance between ship B
and Lighthouse Rock at time t, as shown in the figure below.
Let θ be the angle shown in the
figure. Find the rate of change of θ,
in radians per hour, when x = 4 km
and y = 3 km
At x = 4 and y = 3, secθ = 5/4
tan  
y
x
d
sec 

dt
2
x
dy
dx
y
dt
dt
x2
dy
dx
y
d
 dt 2 dt cos 2 
dt
x
d 17 rad

hr
dt
5
x
Mr. G
HW 1
Section 2.6
Page 154-157
Ex:(9) 27, 31, 32-36, 45, 54
Mr. G