Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Related Rates refers to variables which are related in a known way that
also change in time.
Static Case:
Word Problem (Elementary Algebra): If Yancey picked 30 more apples
than Uri and Xan picked twice as many as Yancy and Uri combined, how
many did each pick if there were 210 apples picked.
Let y = #apples picked by Yancey
u = #apples picked by Uri
x = #apples picked by Xan
1
Chapter3.8.notebook
March 03, 2016
Dynamic Case: How they change in time (speed)
Word Problem (Calculus) If Yancey picks apples three times faster than
Uri and Uri picks apples twice as fast as Xan, how fast does Yancey pick
relative to Xan?
Let
be the rate Yancey picks relative to Uri.
Let
be the rate Uri picks relative to Xan
Find
the rate at which Yancey picks relative to Xan.
2
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Related Rates refers to variables which are related in a known way that
also change in time.
Example:
1. An oil spill on the surface of the ocean spreads out in time.
2. A 16-ft ladder slips down a wall.
3. A 7x10 meter rectangular water tank gets filled.
4. A sand pile accumulates on the floor from a leaky overhead bin in
such a way that the radius of the pile is always three times its height.
5. From a fixed position of 200m away, you observe the launching of a
hot air balloon.
3
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
What are the variables that are in the 'known' relationship and what is
that relationship?
Example:
1. An oil spill on the surface of the ocean spreads out in time.
2. A 16-ft ladder slips down a wall.
3. A 7x10 meter rectangular water tank gets filled.
4
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
What are the variables that are in the 'known' relationship and what is
that relationship?
Example:
4. A sand pile accumulates on the floor from a leaky overhead bin in
such a way that the radius of the pile is always three times its height.
5. From a fixed position of 200m away, you observe the launching of a
hot air balloon.
5
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
What is the relationship between the rates in which the variables change?
How does that rate change as time goes on?
Procedure: Define the static equation then calculate the derivative with
respect to time, t (rate) using implicit differentiation.
We use implicit differentiation because we are always taking the
derivative with respect to time, t, and we don't know each variables
explicit function of time, only in relation to each other.
1. An oil spill on the surface of the ocean spreads out in time. If the
radius increases at a rate of 30m/hr, how fast is the area increasing when
r = 100m? when r = 200m?
6
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
What is the relationship between the rates in which the variables change?
How does that rate change as time goes on?
Example:
2. A 16-ft ladder slips down a wall. If the base of the ladder is 5ft away
from the wall when it starts to slide and it slides at a rate of 3ft/s, find the
velocity of the top of the ladder as it slides down the wall at t = 1s later, at
t = 2s.
7
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Procedure:
1. Draw a picture
2. State what is given and what you need to find.
3. Construct your 'static' equation.
4. Use implicit differentiation on the 'static' equation to produce the Related Rates
Equation.
5. Plug in your know values and solve for the unknown variable.
Example:
2. A 16-ft ladder slips down a wall. If the base of the ladder is 5ft away from the
wall when it starts to slide and it slides at a rate of 3ft/s, find the velocity of the
top of the ladder as it slides down the wall at t = 1s later, at t = 2s.
8
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Procedure:
1. Draw a picture
2. State what is given and what you need to find.
3. Construct your 'static' equation.
4. Use implicit differentiation on the 'static' equation to produce the Related Rates
Equation.
5. Plug in your know values and solve for the unknown variable.
Example:
3. A 7x10 meter rectangular water tank gets filled. Water pours at a rate of
3m3/min into the tank. How fast is the level of the tank rising at 1 min, at 2 min?
9
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Procedure:
1. Draw a picture
2. State what is given and what you need to find.
3. Construct your 'static' equation.
4. Use implicit differentiation on the 'static' equation to produce the Related Rates
Equation.
5. Plug in your know values and solve for the unknown variable.
Example:
4. A sand pile accumulates on the floor from a leaky overhead bin in such a way
that the radius of the pile is always three times its height. If the sand falls at a
constant rate of 120 ft3/min, how fast is the height changing when h = 10ft, h =
20ft?
10
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Procedure:
1. Draw a picture
2. State what is given and what you need to find.
3. Construct your 'static' equation.
4. Use implicit differentiation on the 'static' equation to produce the Related Rates
Equation.
5. Plug in your know values and solve for the unknown variable.
Example:
5. From a fixed position of 200m away, you observe the launching of a hot air
balloon. If the balloon is rising at a speed of 4m/s, how fast is the angle in which
you observe the balloon changing 30s after launch, 40s after launch?
11
Chapter3.8.notebook
March 03, 2016
Chapter 3.8 Related Rates
Procedure:
1. Draw a picture
2. State what is given and what you need to find.
3. Construct your 'static' equation.
4. Use implicit differentiation on the 'static' equation to produce the Related Rates
Equation.
5. Plug in your know values and solve for the unknown variable.
Example:
5. From a fixed position of 200m away, you observe the launching of a hot air
balloon. If the balloon is rising at a speed of 4m/s, how fast is the angle in which
you observe the balloon changing 30s after launch, 40s after launch?
12