1. Differentiation: Related Rates – Day 1
Chapter 2
Differentiation:
Related Rates – Day 1

2. Related Rates
When we talk about a quantity change with respect to time (whether something is increasing, decreasing, growing, shrinking, etc.), we are basically talking about the derivative of that quantity with respect to time.
For example, let’s say little Johnny is growing at a rate of 2 inches per year. We can write this rate mathematically:
Basically, this tells me that the change in height with respect to time is 2 in. per year.

3. Related Rates
Determine the mathematical expressions for each of the following:
The radius of a circle is increasing at 4 ft/min
The volume of a cone is decreasing at 2 in3/second
The account is shrinking by $50,000/day

4. Related Rates
Often times, we want to know how one variable is related to another variable with respect to time. In other words, we want to know how rates are related; thus, we study related rates!!!
In order to determine the relationship among rates, we need to differentiate with respect to time; in other words, we need to be able to determine:

5. Related Rates
For example, let’s say that a rectangle is 10in x 6in whose sides are changing. The formulas for both the perimeter and area in terms of l and w are:
The rate of change for the perimeter is:
The rate of change for the area is:

6. Related Rates
For example, let’s say that a rectangle is 10in x 6in whose sides are changing.
The formulas for the rates of change if the length and width are increasing at a rate of 2 in/s are:
Given this information, we can determine the rate of change of perimeter:

7. Related Rates
For example, let’s say that a rectangle is 10in x 6in whose sides are changing.
The formulas for the rates of change if the length and width are increasing at a rate of 2 in/s are:
Given this information, we can also determine the rate of change of area:

8. Related Rates
Related rates can be difficult for some to master, so I’ve come up with 5 steps to help you solve these problems:
Make a sketch of the situation, introducing all known and unknown variables and information
Identify the given rates of change and determine the rate of change to be found (all rates of change should be written as derivatives)
Write the equation that relates the variables. You may have to write several equations.
Differentiate with respect to time, t
Substitute known information to solve for the desired rate of change.

9. Related Rates
Note: It is a common MISTAKE to try to substitute known information before differentiating implicitly. Do not try to substitute the known information until you have completed the implicit differentiation!!!

10. Example 1:
An oil tanker hits an iceberg and starts to spill oil creating a circular region. The oil is rapidly spilling out of the tanker and the radius of the circle increases at rate of 2 ft/s. How fast is the area of the oil increasing when the radius is 60 ft?
Step 1: Make a sketch
At this exact point in time, the radius is 60 ft (r = 60 ft).

11. Example 1:
An oil tanker hits an iceberg and starts to spill oil creating a circular region. The oil is rapidly spilling out of the tanker and the radius of the circle increases at rate of 2 ft/s. How fast is the area of the oil increasing when the radius is 60 ft?
Step 2: Identify given rate(s) and rate(s) to be found

12. Example 1:
An oil tanker hits an iceberg and starts to spill oil creating a circular region. The oil is rapidly spilling out of the tanker and the radius of the circle increases at rate of 2 ft/s. How fast is the area of the oil increasing when the radius is 60 ft?
Step 3: Write the equation(s) that relate the variables
The area of a circle formula:

13. Example 1:
An oil tanker hits an iceberg and starts to spill oil creating a circular region. The oil is rapidly spilling out of the tanker and the radius of the circle increases at rate of 2 ft/s. How fast is the area of the oil increasing when the radius is 60 ft?
Step 4: Differentiate with respect to time

14. Example 1:
An oil tanker hits an iceberg and starts to spill oil creating a circular region. The oil is rapidly spilling out of the tanker and the radius of the circle increases at rate of 2 ft/s. How fast is the area of the oil increasing when the radius is 60 ft?
Step 5: Substitute and solve for the desired rate

15. Example 2:
A ladder 20 ft. long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft. per second, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground?
Step 1: Make a sketch
At this exact point in time, the length of the ladder is 20 ft and the height of the ladder is 12 ft (y = 12 ft). So, by the Pythagorean Theorem, the base must be 16 ft away from the wall (x = 16 ft).

16. Example 2:
A ladder 20 ft. long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft. per second, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground?
Step 2: Identify given rate(s) and rate(s) to be found

17. Example 2:
A ladder 20 ft. long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft. per second, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground?
Step 3: Write the equation(s) that relate the variables
The Pythagorean Theorem will related the variables:

18. Example 2:
A ladder 20 ft. long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft. per second, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground?
Step 4: Differentiate with respect to time

19. Example 2:
A ladder 20 ft. long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft. per second, how fast is the ladder sliding down the wall when the top of the ladder is 12 feet above the ground?
Step 5: Substitute and solve for the desired rate

20. Homework Related Rates Read Sections 2.6 P. 154: 1 – 35 (odd) and 36
WS: Related Rates Homework
WS: Related Rates Turvey
WS: Related Rates Problem Set

21. Differentiation: Related Rates – Day 2
Chapter 2
Differentiation:
Related Rates – Day 2

22. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 1: Make a sketch
At this exact point in time, the height of the rocket is 4000 ft (y = 4000 ft) and the height of the horizontal distance from the rocket to the camera is 3000 ft (this is constant, so we don’t give it a variable!) θ

23. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 2: Identify given rate(s) and rate(s) to be found

24. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 3: Write the equation(s) that relate the variables
The equation that will relate y with θ is tan θ :

25. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 4: Differentiate with respect to time

26. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 5: Substitute and solve for the desired rate
Before we can substitute and solve, we need to know dy/dt and θ. We were given dy/dt, but we still need to solve for θ at this point in time.

27. Example 3:
A space shuttle is launched and is rising at a rate of 880 ft/s when it is 4000 ft up. If a camera is mounted on the ground 3000 feet away from where the shuttle is launched, how fast must a camera angle change at that instant in order to keep the shuttle in sight?
Step 5: Substitute and solve for the desired rate

28. Example 4: Step 1: Make a sketch
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep. Note 1: 1 gallon = cu ft.; Note 2: The water fills rapidly at first, but slows down as the cone gets wider.
Step 1: Make a sketch h
At this exact point in time, the height of the water is 3 ft (h = 3 ft); however, the height of the entire cone is 12 ft. and the radius of the entire cone is 6 ft.

29. Example 4: Step 2: Identify given rate(s) and rate(s) to be found
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 2: Identify given rate(s) and rate(s) to be found
NOTE: There are too many unknown variables!!! So, we will eventually need to eliminate one!!!

30. Example 4: Step 3: Write the equation(s) that relate the variables
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 3: Write the equation(s) that relate the variables
The equation that will relate r and h with V is:
At this point, we need to eliminate one of the variables; the rate of volume is given and we are looking for the rate that water rises (height), so we need to try to eliminate the radius!

31. Example 4:
In order to eliminate the radius from the equation (i.e. write volume using only height), we must go back to geometry and consider similar triangles: h h
Set up a ratio of similar triangles and solve for r.

32. Example 4: Step 3: Write the equation(s) that relate the variables
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 3: Write the equation(s) that relate the variables
The equation that will relate h with V is:
Note: When we complete step 3, the equation should ONLY include the variables for which the rates are GIVEN and the variable whose rate we are finding!!!

33. Example 4: Step 4: Differentiate with respect to time
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 4: Differentiate with respect to time

34. Example 4: Step 5: Substitute and solve for the desired rate
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 5: Substitute and solve for the desired rate
Before we can solve, we need to convert from gallons to feet:

35. Example 4: Step 5: Substitute and solve for the desired rate
You are trying to fill a tank in the shape of a cone with an altitude of 12 ft and a radius of 6 ft with water. The water is getting pumped into the tank at a rate of 10 gallons per minute. Find the rate at which the water rises when the water is 3 feet deep.
Step 5: Substitute and solve for the desired rate

36. Homework Related Rates Read Sections 2.6 P. 154: 1 – 35 (odd) and 36
WS: Related Rates Homework
WS: Related Rates Turvey
WS: Related Rates Problem Set

37. Differentiation: Related Rates – Day 3
Chapter 2
Differentiation:
Related Rates – Day 3

38. Example 5: Step 1: Make a sketch
An 11 foot by 20 foot swimming pool is being filled at a constant rate of 5 ft3 per minute. The pool has a depth of 4 feet in the shallow end and 9 feet in the deep end. (Assume the pool’s depth increases constantly between the shallow end and the deep end.) How fast is the height of the water increasing when the water has filled 2 feet of the deep end?
Step 1: Make a sketch
At this exact point in time, the height of the water is 2 ft (h = 2 ft)

39. Example 5: Step 2: Identify given rate(s) and rate(s) to be found
An 11 foot by 20 foot swimming pool is being filled at a constant rate of 5 ft3 per minute. The pool has a depth of 4 feet in the shallow end and 9 feet in the deep end. (Assume the pool’s depth increases constantly between the shallow end and the deep end.) How fast is the height of the water increasing when the water has filled 2 feet of the deep end?
Step 2: Identify given rate(s) and rate(s) to be found
NOTE: There are too many unknown variables. So, we will eventually need to eliminate one.

40. Example 5: Step 3: Write the equation(s) that relate the variables
This is a tough one!
The shape of the pool is a 3-dimensional trapezoid or a trapezoidal prism. It will be very difficult to work with the equation:
So what do we do?
We need to look at the shape in another way.
Well, if we think about how water rises in the shape, it starts at the bottom of the deep end.

41. Example 5: Step 3: Write the equation(s) that relate the variables
While the water is rising from 0 to 5 feet (since 9ft – 4ft = 5 ft), our 3-dimensional shape is a triangular prism – a much easier shape to solve:
This part is superfluous
Now finding the volume of a triangular prism is much easier:
Width is constant, so:

42. Example 5:
There are still too many variables. We know dV/dt and we are looking for dh/dt, so we should try to eliminate b (in other words, we need to write the equation for volume in terms of height alone).
Set up a ratio of similar triangles and solve for b.

43. Example 5: Step 4: Differentiate with respect to time
An 11 foot by 20 foot swimming pool is being filled at a constant rate of 5 ft3 per minute. The pool has a depth of 4 feet in the shallow end and 9 feet in the deep end. (Assume the pool’s depth increases constantly between the shallow end and the deep end.) How fast is the height of the water increasing when the water has filled 2 feet of the deep end?
Step 4: Differentiate with respect to time

44. Example 5: Step 5: Substitute and solve for the desired rate
An 11 foot by 20 foot swimming pool is being filled at a constant rate of 5 ft3 per minute. The pool has a depth of 4 feet in the shallow end and 9 feet in the deep end. (Assume the pool’s depth increases constantly between the shallow end and the deep end.) How fast is the height of the water increasing when the water has filled 2 feet of the deep end?
Step 5: Substitute and solve for the desired rate

45. Homework Related Rates Read Sections 2.6 P. 154: 1 – 35 (odd) and 36
WS: Related Rates Homework
WS: Related Rates Turvey
WS: Related Rates Problem Set