3.11 Related Rates
Mon Nov 10
• Do Now
• Differentiate implicitly in terms of t
• 1) A = 1 bh
2
• 2)
a +b =c
2
2
2
Related Rates
• When we use implicit differentiation, we
obtain dy/dx, or the change of y in terms of x.
• In many real life situations, each quantity in
an equation changes with time (or another
variable)
• In this case, any derivative we find is called a
related rate, since each rate in the derivative
is related to each other
Related Rates Steps
• 1) Make a simple sketch, if possible
• 2) Identify what rate you are looking for
• 3) Set up an equation relating ALL of the
relevant quantities
• 4) Differentiate both sides of the equation in
terms of the variable you want
– if you want dv/dt, you differentiate in terms of t
• 5) Substitute in values we know
• 6) Solve for the remaining rate
Ex 1
• A 5-meter ladder leans against a wall. The
bottom of the ladder is 1.5 m from the wall at
time t=0 and slides away from the wall at a
rate of 0.8m/s. Find the velocity of the top of
the ladder at time t=1
Ex 2
• Water pours into a fish tank at a rate of 0.3
m^3 / min. How fast is the water level rising if
the base of the tank is a rectangle of
dimensions 2 x 3 meters?
Ex 3
• Water pours into a conical tank of height 10 m
and radius 4 m at a rate of 6 m^3/min
• A) At what rate is the water level rising when
the level is 5 m high?
• B) As time passes what happens to the rate
at which the water level rises?
Ex 4
• A spy uses a telescope to track a rocket
launched vertically from a launching pad 6km
away. At a certain moment, the angle
between the telescope and ground is equal to
pi/3 and is changing at a rate of 0.9
radians/min. What is the rocket’s velocity at
that moment?
Ex 5
• See book
Closure
• At what rate is the diagonal of a square
increasing if its sides are increasing at a
rate of 2 cm/s?
• HW: p.199 #1-37 every other odd
• Ch 3 Test next week? Mon?
3.11 Related Rates Cont’d
Tues Nov 11
• Do Now
• Air is being pumped into a spherical
balloon at a rate of 5
cm3/min. Determine the rate at which
the radius of the balloon is increasing
when the radius of the balloon is 10 cm.
• (hint: Volume = 4/3 pi x r^3)
HW Review p.199 #1-35
• Probably all of them
More practice
• worksheet
Closure
• Hand in: A 15 foot ladder is resting against
the wall. The bottom is initially x feet away
from the wall and is being pushed towards
the wall at a rate of 0.5 ft/sec. How fast is the
top of the ladder moving up the wall when the
bottom of the ladder is 4 feet from the wall??
(Hint: Use Pythagorean Theorem)
• HW: p.199 #1-35 all other odd
• p.AP3-1 #1-20, 1-4 due Thurs
• P.203 #5-11, 17-25, 29-75 85-115 119-123
due Fri