Related Rates (Section 3.8)
Strategy:
1.
Draw
2.
Label (use
variables except for values that never change)
3.
Know (What do you
know.)
4.
Want (What do you
want to know?)
5.
Relate
(Relationships between variables)
6.
Differentiate
(With respect to t)
7.
Substitute and
Solve
Related Rates Example Problems
1)
A rectangle’s
length increases at the rate of 4 mm/hr and its width decreases at the rate of
2 mm/hr. At what rate is the rectangle’s
area changing when its length is 150 mm and its width is 100 mm?
2)
A particle moves
along the curve 
. As the particle
passes through the point (4,2), its x-coordinate increases at a rate of 3
cm/sec. How fast is the distance from
the particle to the origin changing at this instant?
3)
Two cars start
moving from the same point. One travels
south at 30 mi/hr and the other travels east at 40 mi/hr. At what rate is the distance between the cars
increasing 1 hour later?
4)
A balloon is
rising at a constant speed of 5 ft/sec.
A boy is cycling along a straight road at a speed of 15 ft/sec. When he passes under the balloon, it is 45
feet above him. How fast is the distance
between the by and the balloon increasing 3 seconds later?
5)
(Students Try) A
13 foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house,
the base is moving at a rate of 5ft/sec. How fast is the top of the ladder
sliding down the wall then?
6)
At noon, ship A
is 150 km west of ship B. Ship A is
sailing east at 35 
and ship B is sailing
north at 25 . How fast is the
distance between the ships changing at
7)
A spotlight on
the ground shines on a wall 12 m away.
If a man 2 m tall walks from the spotlight toward the building at a speed
of 1.6 m/s, how fast is the length of his shadow on the building decreasing
when he is 4 m from the building? (#16 pg. 187)
8) A man starts walking north
at 1 ft/sec from a point P. One minute later
a woman starts walking south at 3 ft/sec from a point 400 feet due east of
P. At what rate are the people moving
apart 2 minutes after the man starts walking? (#17pg. 187)
9)
Sand falls from a
conveyor belt at the rate of 
onto the top of a conical pile. The height of the pile is always
three-eighths of the base diameter. How
fast are the (a) height and (b) radius changing when the pile is 4 m high?
10)A
13 foot ladder is leaning against a house when its base starts to slide
away. By the time the base is 12 ft from
the house, the base is moving at a rate of 5ft/sec. At what rate is the angle
between the ladder and the ground changing then?
11)A
particle moves along a parabola 
in the first quadrant
in such a way that its x-coordinate (measured in meters) increases at a steady
rate of 10 m/sec. How fast is the angle
of inclination 
of the line joining
the particle to the origin changing when x = 3 m? NOTE: See “Angle of Inclination” under “Helpful Terms and Formulas” below.
12)The law of cosines relating
the lengths of the three sides of a triangle is 
where is the angle opposite
side c. If sides a = 1 and
b = 2 are of fixed length, and the angle is increasing at the
rate of 0.2 rad/min, find the rate at which c is increasing at the instant when
c = 
.
Helpful terms and formulas:
·
Distance between
two points 
and 
is D= 
.
·
Angle of
Elevation: The angle between the line of sight and the horizontal.
·
Angle of
Depression: The angle between the horizontal and the line of sight.
·
Angle of
Inclination of a Line: the angle measured counterclockwise from the positive
side of the x-axis to the line.
·
HW #: Note: If C=
rate at which water is pumped in (Or call it 
if you like) then 
, which means that C = 
.
·
HW #: Volume of a
Triangular Prism: 
V= Area of Base of
Prism * Height of Prism