Related rates

1
Related rates
2
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
3
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
z
y
x
4
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
z
y
x
Differentiate w.r.t. time
5
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
z
y
x
When
6
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
z
y
x
When
7
A boat is pulled into a dock by a rope attached
to the bow of the boat and passing through a
pulley on the dock that is 1 metre higher than
the bow of the boat. If the rope is pulled in at
a rate of 1 m/s, how fast is the boat approaching
the dock when it is 8 m from the dock?
z
y
x
The ve sign means that x is decreasing which is
what we expect
8
The height of a cone increases at a rate of 2 cm
per second but the length of the slant side
remains constant at 9 cm. At what rate is the
volume changing when h 4 cm?
9
Step 1 Draw a diagram

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

10
Step 2 Write the rate(s) given

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

11
Step 3 Write the rate equation you need to solve

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

12
Step 4 Focus on what is missing and write an
equation for this in one variable

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

13
Step 5 Differentiate and THEN substitute

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

14
Step 6 Substitute and calculate

• The height of a cone increases at a rate of 2 cm
  per second but the length of the slant side
  remains constant at 9 cm. At what rate is the
  volume changing when h 4 cm?

Check units
15

• Water is poured at the rate of 0.01 litre/second
  into the conical container shown below. Assume
  the container is empty at the start of the
  experiment (t 0), find the rate of change of h
  the height of the water in the container at t 3
  seconds.

16
Write down the information you have been given

• Water is poured at the rate of 0.01 litre/second
  into the conical container shown below. Assume
  the container is empty at the start of the
  experiment (t 0), find the rate of change of h
  the height of the water in the container at t 3
  seconds.

17
Write down the information you have been given

• Water is poured at the rate of 0.01 litre/second
  into the conical container shown below. Assume
  the container is empty at the start of the
  experiment (t 0), find the rate of change of h
  the height of the water in the container at t 3
  seconds.

18
Write down the information you have been given

• Water is poured at the rate of 0.01 litre/second
  into the conical container shown below. Assume
  the container is empty at the start of the
  experiment (t 0), find the rate of change of h
  the height of the water in the container at t 3
  seconds.

19
Split the derivative
20
Write an equation for volume in terms of height
21
Find dV/dh
22
We need to find h when t 3 secs
23
Substitute
24
On a certain clock the minute hand is 4 cm long
and the hour hand is 3 cm long. How fast is the
distance between the tips of the hands changing
at 9 oclock?
25
Step 1 Draw a diagram

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
26
Step 1 Write the equation

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
27
Step 2 Differentiate w.r.t. t

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
28
Step 3 Find d?/dt

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
29
Step 4 Find x when ? p/2

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
30
Step 5 Substitute

• On a certain clock the minute hand is 4 cm long
  and the hour hand is 3 cm long. How fast is the
  distance between the tips of the hands changing
  at 9 oclock?

3
x
?
4
31
Scholarship 2009

• A conical perfume bottle tapers to a point at the
  top. The bottle has been left open, and the rate
  of evaporation of the perfume is proportional to
  the exposed surface area of the remaining
  perfume. The bottle begins 7/8 full, and the
  perfume takes 10 days to evaporate completely.
  When is the bottle half-full?

32
Label the diagram

• A conical perfume bottle tapers to a point at the
  top. The bottle has been left open, and the rate
  of evaporation of the perfume is proportional to
  the exposed surface area of the remaining
  perfume. The bottle begins 7/8 full, and the
  perfume takes 10 days to evaporate completely.
  When is the bottle half-full?

r
H
h
R
33
Set up the equation

• A conical perfume bottle tapers to a point at the
  top. The bottle has been left open, and the rate
  of evaporation of the perfume is proportional to
  the exposed surface area of the remaining
  perfume. The bottle begins 7/8 full, and the
  perfume takes 10 days to evaporate completely.
  When is the bottle half-full?

r
H
h
R