AP CALCULUS - AB
Section Number:
2.6
LECTURE NOTES
Topics: Related Rates (Day 1)
- The Language of “Related Rates” Problems
- Circle and Pythagorean Theorem Problem
Day
Calculus is “the mathematics of change.” Words that mean change are: increasing, decreasing, growing,
shrinking, etc. Change occurs over time. So, when we talk about how a quantity changes, we are talking
about the derivative of that quantity with respect to time.
Example 1:
Write the following statements mathematically.
a. John is growing at the rate of 3 inches/year
b. My mutual fund is shrinking by 4 cents/day
c. The radius of a circle is increasing by 4 ft/hr
d. The volume of a cone is decreasing by 2 in3/sec
Example 2: Rectangle A rectangle is 10 inches by 6 inches whose sides are changing. Write formulas for
both the perimeter and area and how fast each is changing in terms of l and w.
Perimeter
Change of Perimeter
Area
Change of Area
a. its length and width are increasing at the rate of 2 inches/sec
Change of Perimeter
Change of Area
b. its length and width are decreasing at the rate of 2 inches/sec
Change of Perimeter
Change of Area
c. its length is increasing at 3 inches/sec and its width is decreasing at 3 inches/sec
Change of Perimeter
Change of Area
d. its length is decreasing at 2 inches/sec and its width is increasing at 0.5 inches/sec
Change of Perimeter
Change of Area
Example 3: Right Triangle A right triangle has sides of 30 and 40 inches whose sides are changing. Write
formulas for the area of the triangle and the hypotenuse of the triangle and how fast they are each changing.
Area
Change of Area
Hypotenuse
Change of Hypotenuse
a. the short side is increasing at 3 in./sec and the long side is increasing at 5 in/sec
Change of Area
Change of Hypotenuse
b. the short side is increasing at 3 in./sec and the long side is decreasing at 5 in/sec
Change of Area
Change of Hypotenuse
Example 4: Cylinder A right circular cylinder has a height of 10 feet and a radius of 8 feet whose
dimensions are changing. Write formulas for the volume and surface area of the cylinder and the rates at which
they each change.
Volume
Change of Volume
Surface Area
Change of Surface Area
a. the radius is growing at 2 feet/min and the height is shrinking at 3 feet/min
Change of Volume
Change of Surface Area
b. the radius is decreasing at 4 feet/min and the height is increasing at 2 feet/min
Change of Volume
Change of Surface Area
To solve related rates problems, you need a strategy. Related rates problems always can be recognized by the
words “increasing, decreasing, growing, shrinking, and changing.” Follow these guidelines.
SOLVING RELATES RATES PROBLEMS
Step 1: Make a sketch and label all known and unknown quantities.
The derivative with respect to time is the rate.
Step 2: Write down the sides and the rates that are given. Write down the rate you are trying to find.
Step 3: Find an equation that ties your variables together.
If the problem mentions area, you will most likely write an area formula.
If the problem mentions volume, you will most likely write a volume formula.
If the problem mentions lengths of sides and the figure is a right triangle,
you will most likely use the Pythagorean Theorem as your equation.
If the problem mentions angles measures and the figure is a right triangle,
you will most likely use a trigonometric ratio as your equation.
Step 5: Differentiate your equation with respect to time.
Step 6: Substitute the known quantities and solve
Example 5:
A Rock in a Pond
A rock is dropped in a pond. The radius of the ripple increases at 4 ft/sec. At what rate is the
area of the circle expanding when the radius is 12 feet?
Example 6:
Oil Tanker Spill
An oil tanker spills oil that spreads in a circular pattern whose radius increases at the rate
of 50 feet/min. How fast is the circumference the spill increasing when the radius of the spill is
20 feet?
Example 7:
Oil Tanker Spill 2
An oil tanker spills oil that spreads in a circular pattern whose radius increases at the rate
of 50 feet/min. How fast is the area of the spill increasing when the radius of the spill is 20 feet?
Example 8:
Balloon
The volume of a spherical balloon is increasing at a rate of 15 cubic inches / minute. At what
rate is the radius changing when the radius is 6 inches?
Example 9:
A Falling Ladder
A 13 foot ladder leans against a vertical wall. If the lower end of the ladder is pulled away at the
rate 2 feet per second, how fast is the top of the ladder coming down the wall at
a.) the instant the top of the ladder is 12 feet above the ground and
b.)…5 feet above the ground?
AP CALCULUS - AB
Section Number:
2.6
Example 10:
LECTURE NOTES
Topics: Related Rates (Day 2)
- Pythagorean Theorem Problem 2
- Trigonometric Ratio Problem
- Cone Problems
Day:
A Rotating Camera
A camera is mounted 3,000 feet from the space shuttle launching pad. The camera
needs to pivot as the shuttle is launched and needs to keep the shuttle in focus. If the
shuttle is rising vertically at 800 feet/sec when it is 4,000 feet high, how fast is the
camera-to-shuttle distance changing?
Consider the same problem above. How fast is the angle of elevation of the camera
changing at that moment when the shuttle is 4,000 feet high?
Example 11:
A Pile of Sand (Cone Problem 1)
Sand is poured on a beach creating a cone whose radius is always equal to twice its height. If
the sand is poured at the rate of 20 in3/sec, how fast is the height of the conical pile changing at
1
the time the height is a.) 2 inches and b.) 6 inches ? (The volume of a cone is V  r 2 h .)
3
Example 12: A Draining Tank of Water (Cone Problem 2)
Water is draining from a conical tank (with vertex down) at the rate of 2 meters3/sec. The tank
is 16 meters high and its top radius is 4 meters. How fast is the water level falling when the
water level is a.) 12 meters high and b.) 2 meters high ?
AP CALCULUS - AB
Section Number:
2.6
LECTURE NOTES
Topics: Related Rates (Day 3)
- Shadow Problem
- More Challenging Trigonometric Problem
Day
Example 13: “Me and My Shadow…”
Suppose a 6 foot tall person is 12 ft away from an 18-feet tall lamppost. If the person is
moving away at a rate of 2 ft/sec, at what rate is the length of the shadow changing?
At what rate is the tip of the shadow changing?
Example 14: Cars on the Road
Two cars are riding on roads that meet at a 60° angle. Car A is 3 miles from the intersection
traveling at 40 mph and car B is 2 miles away from the intersection traveling at 50 mph. How
fast are the two cars separating if a.) they are both traveling away from the intersection and
b.) car A is traveling away from the intersection and Car B is traveling towards it?
Example 15: Coffee Cup
Coffee is being poured into a conical cup. Find the rate of change of the volume of the cone if
the radius is increasing at 2 in / min and when h = 3r and when r = 6 inches.
Example 16: Water Cup
Water is poured into a conical paper cup at the rate of 4 cubic centimeters per second. The
paper cup measures 8 cm tall and the top has a radius of 5
a.) Use similar triangles to express the height h in terms of the radius r. Be sure to solve your
equation for r.
a.) Determine how fast the water level is rising at the instant the water is 3 cm deep.