• To use differentiation to solve problems dealing with optimization, rates of change, and distance-velocity-acceleration. Supported by learning objectives 1 through 3.
| F.1 |
| F.2 |
| F.3 |
| F.4 |
Objective
F.1 Back to Top
To solve problems that involve rates of change.
Instructional Notes
In units D and E, students have encountered occasional problems involving rates of change while developing skills in differentiation and curve sketching. This unit gives the students an opportunity to review and extend their problem-solving skills in this area. In this unit, we want students to be able to interpret the slope of a tangent line as an instantaneous rate of change and the slope of a secant line as an average rate of change. It is most important that students provide units along with their answers. Refer students back to D.4 in which they were first introduced to the slope of the tangent line as a rate of change.
Introduce this objective with the following example.
Example: During a 24-hour time period, the temperature in degrees Celsius in Our Town was given by the function 
where x is the number of hours that have passed.
(a) Draw a graph of the function for 
.
(b) Find the temperature after 5 hours and after 15 hours.
(c) Find the average rate of change in temperature during those 10 hours. Interpret this graphically.
(d) Find the instantaneous rate of change in the temperature after 5 hours and after 15 hours.
Solution:
(a) The graph appears below. Students could create a table of values with a calculator and then plot the graph, or students could use a graphing utility and then copy the graph from the utility.
(b) 
and 
.
(c) The average rate of change in temperature during those 10 hours could be determined by finding: 
(d) Graphically, this is the slope of the secant line connecting points A and B.
(e) To find the instantaneous rate of change after 5 hours, find the slope of the tangent line at point A . (If we find the average rate of change over shorter and shorter time intervals, the secant line approaches the tangent line.)
Now 
, thus 
and 
.
At the 5-hour mark, the temperature is increasing at a rate of 2.15 degrees per hour and at the 15-hour mark, the temperature is decreasing at a rate of .05 degrees per hour.
Examples/Activities
1. A weather balloon that contains 
of helium springs a leak and empties in 60 seconds. Suppose that the volume of helium in the balloon as a function of time is given by 
. Find:
(a) the average rate of change of volume during the 60 seconds it takes to empty.
(b) the volume of the balloon after 40 seconds.
(c) the instantaneous rate of change of the volume at 40 seconds.
2. If 
, and x changes from 1 to 5, what is the average change in y ? What is the instantaneous rate of change in y when 
?
3. If a rock is thrown into the water, circular ripples begin to expand outward from the point where the rock entered the water. The area of the circular ripple is given by the function 
.
(a) What is the average change in area as r changes from 3 m to 7 m?
(b) What is the instantaneous rate of change in area with respect to the radius when the radius is 6 m?
4. Suppose that the number of people in your school that hear a rumour about you x hours after it is started is given by the function 
.
(a) What is the average rate at which the rumour is spreading between 
and 
?
(b) What is the instantaneous rate at which the rumour is spreading after 4 hours?
(c) Is the rate at which the rumour is spreading increasing or decreasing? Hint: Consider the graph of the function.
5. The value of a new car x years after it is purchased is given by the function 
.
(a) What is the value of the car when it is new?
(b) What is the value of the car when it is five years old?
(c) What is the average yearly drop in price during those five years?
(d) What is the instantaneous rate of change in price after 4 years (that is when 
)?
6. If gasoline costs $.80/L and you drive 24000 km per year, then your annual cost for gasoline will be given by the function 
where x is the number of km/L that your car can travel.
(a) Find your annual gasoline costs if your car gets 3 km/L.
(b) Find your annual gasoline costs if your car gets 8 km/L.
(c) What is the average rate of change in your annual gasoline cost as x changes from 3 to 8?
(d) What is the instantaneous rate of change in your annual fuel costs when 
?
Suggestions/Extensions
Have the students discuss situations familiar to them that involve instantaneous and/or average rates of change. Students can also create problems for the class to solve based on these situations.
Objective
F.2 Back to Top
To solve a wide variety of optimization problems.
To pose questions and seek clarification from peers, the teacher, and other sources (CCT).
Instructional Notes
One of the most powerful and intriguing applications of calculus is seen in the solving of optimization problems. Every calculus resource book will contain these although it may be necessary to seek out several such books in order to find a sufficient number of problems that students can handle without becoming frustrated. The typical university calculus text has only a few problems at an easier level.
An introductory problem is found in an interactive format in the software Journey Through Calculus ( see the Calculus Bibliography). Be aware that this is a challenging problem and it might be appropriate to provide some simpler problems first so students experience success early on.
The problem concerns the dilemma of a lifeguard who notices a swimmer in distress. The swimmer is 50 m from the shoreline. The lifeguard is 150 m from the point on the shore that is directly opposite the swimmer. The guard can run at a speed of 6 m/s and swim at a speed of 3 m/s. What path (down the beach and through the water) should the guard follow to get to the swimmer in the least time?
Posing this problem and letting the students “go at it” unassisted for quite a while will pay several dividends. Most students will begin to solve the problem by trial and error. By using actual values, students become familiar with the problem, and are thus better equipped to form a variable expression for the time taken. Hopefully, their experimental work will lead to questions (CCT):
• Can I obtain an expression for the time taken in terms of the point at which the lifeguard enters the water? If I get that expression, what could I do with it? How does calculus relate to this situation?
Working through the entire solution using the two most likely assignments for x will show how different assignments lead to different levels of difficulty in differentiation and solving the resulting equation.
The equation on the left is easier to work with, and the rescue time is minimized if the lifeguard runs 121.13 m before entering the water. Drawing the function on the graphing calculator and using the Calculate Minimum option will help confirm the solution. Students can see the graph of the function and realize they should differentiate to find the local minimum. Remind the students that if they have not seen the graph of the function, they should not assume that the absolute minimum occurs at a local minimum. They should really be checking to find the values of 
and 
to see if they give a shorter time. Noting that the domain of the function that gives the lifeguard's time is 
is also extremely critical. E.2 is relevant here.
The introductory problem is illustrative of the power of calculus and the messy process that is sometimes encountered.
Next, move to some examples that are not as difficult. It is best if students experience initial success. Spending one class period on optimization problems related to numbers and another class period on problems related to geometric figures is beneficial before expecting students to solve the type of problem used in the introduction. You might create a take-home project containing 4 or 5 of the more difficult problems.
Here are some tips to pass along at the appropriate moment. There is little point in running through all of these at once.
• Read the question carefully. What is the quantity that you are trying to maximize or minimize?
• Find an expression for that quantity in terms of the variables involved. If you cannot come up with the expression, experiment with guesses until you can see a pattern. Find a relationship among the variables so that you can write an expression for the quantity in terms of only one variable. If you are having trouble writing the expression in terms of one variable, look for relationships among the variables. If you are working with a geometric figure, the Theorem of Pythagoras or similar triangles often lead to such a relationship.
• If the problem concerns a geometric figure, draw a very well labelled diagram. This will often help to create the desired expression.
• When you have created the expression, find its derivative and determine the critical numbers.
• Evaluate the expression at each of the critical numbers as well as at the endpoints of the interval of the domain for the function. If the domain of the expression is all real numbers, consider what happens to the value of the expression for very large positive and very large negative values of the variable. Select the critical number(s) that optimize(s) the quantity.
• Conclude in a sentence.
There are many classic problems that appear in calculus books. Two that you should consider having the students solve follow.
1. A rectangular piece of paper measures 20 cm by 28 cm. Equally sized squares are to be cut out from each corner of the paper and the remaining flaps are to be folded up to make an open-topped box. Find the dimensions of the square that should be cut out to maximize the volume of the box.
2. A metal cylindrical pop can is to be constructed to have a known volume, V . What is the ratio of the diameter of the can to the height of the can if the amount of metal used in the construction of the can is to be a minimum? Assume that the metal used is of uniform thickness.
A few more classic problems follow.
1. A Norman window, one that is formed by placing a semicircle on top of a rectangle, still remains a popular architectural feature. If the perimeter of the window is 300 cm, find the radius of the semicircle that will maximize the window's area (let in the most light).
2. A piece of wire 48 cm long is cut into two pieces. One of the pieces is bent to form a circle while the other is bent to form a square. Find the dimensions of the square and the radius of the circle so that the total combined area will be a minimum.
3. Find the dimensions of the cylinder of largest volume that can be inscribed in a sphere of radius 27 cm.
Examples/Activities
Have students begin with optimization problems related to numbers.
1. One number is 4 larger than another. How must they be chosen in order to minimize their product? What is that minimum product?
You might want to also ask the following questions.
How must they be chosen to maximize their product? What is that maximum product?
Note that the last two questions have no answer. Although you do not want to do this to the students regularly, they need to recognize that the critical number is just one place to look for an absolute extrema. They must also consider the endpoints of the domain of the problem. In this case, no restriction is given on the numbers other than one must be 4 larger than another. Thus, clearly 1000 and 1004 give a large product, but not as large as 10000 and 10004.
2. Two numbers have a sum of 4. How must they be chosen in order to maximize their product? What is that maximum product?
3. Two positive numbers have a product of 4. How should they be chosen in order to minimize their sum?
4. What number exceeds its square by the greatest amount? What is that amount?
5. What number exceeds its square root by the least amount? What is that amount?
6. Two positive numbers must have a sum of 15. How should they be chosen if the product of the square of one of them with the cube of the other is to be a maximum? What is that maximum product?
7. Two positive numbers have a product of 9. How should they be chosen so that the sum of their squares will be a minimum? What is that minimum?
Note: In addition, introduce some problems about numbers that are similar to the ones above but require students to work with an abstract constant. Consider the following.
8. Find two positive numbers whose sum is k , if their product is to be a maximum. What is the ratio of one number to the other?
9. Two positive numbers have a sum of k . How should the numbers be chosen if the sum of their squares is to be a minimum? What is that minimum?
After working on number problems, might switch to problems about geometric figures such as the following “classic” problems.
1. Sadie has 60 m of fencing that she plans to use to enclose a rectangular garden plot. Find the dimensions of the plot that will maximize the area. What is the maximum area?
2. Sadie has 60 m of fencing that she plans to use to enclose a rectangular garden plot. One side of the garden will be against the barn, so she does not need to put a fence on that side. Find the dimensions of the plot that will maximize the area. What is the maximum area?
3. Sadie has 60 m of fencing that she plans to use to enclose a rectangular garden plot as well as subdivide the garden into two rectangles – see the figure. Find the dimensions that will maximize the area. What is the maximum area?
4. What are the dimensions of the rectangle of the largest area that has a diagonal of length 60 m?
5. What is the minimum perimeter that a rectangle can have and yet contain an area of 60 
?
6. What are the dimensions of the largest rectangle that can be inscribed in a semi-circle of radius 60 m?
7. What are the dimensions of the largest rectangle that can be inscribed in a right triangle with base 60 m and height 40 m?
8. Find the dimensions of the isosceles triangle of greatest area having a perimeter of 60 m.
9. An eaves trough has a cross section that forms an isosceles trapezoid. If the two legs and the short base of the trapezoid are each 10 cm, find the distance across the top of the trapezoid that will maximize the area of the trapezoid and thus the carrying capacity of the eaves trough.
After tackling problems similar to those above, select one or two of them to be done again, replacing the number 60 by the constant k .
Some three-dimensional problems could be tackled next include:
1. A square piece of tin, 30 cm by 30 cm is to be made into an open-topped box by cutting out equal squares from each corner and folding up the flaps. Find the dimensions of the cut-out square if the volume of the box is to be a maximum.
2. A rectangular box-shaped garbage can with a square base and an open top is to be constructed using exactly 2700
of material. Find the dimensions of the box that will provide the greatest possible volume.
3. Find the dimensions of the cylinder of greatest volume that can be inscribed in a cone whose radius is 30 cm and whose height is 40 cm.
Examples/Activities
The sample questions provided earlier are classic and traditional. What plans do your students have for the future? Collect problems from the different resources that relate to economics, medicine, pharmacy, biology, engineering, etc. and let the students specialize in their area of interest. Make a start on this collection and add to it over the years.
Problems from some of these areas have appeared in this guide in unit E. A few more follow.
1. The strength of a beam varies jointly as its width and the square of its height. Find the dimensions of the strongest beam that can be cut from a log of diameter 30 cm.
2. An oil well has been discovered offshore at W, 200 m from S, the nearest point on the shoreline. Town T is located 1000 m along the shore from point S. A pipeline must be installed underwater from W to V and then along the shoreline from V to T. If it costs $500/m to run the pipe underwater and $200/m to run the pipe along the shore, how far from S should V be located to minimize the total cost of the pipeline?
3. The owner of a condominium complex has 45 units which will be occupied if the rent charged is $600 per month. The owner estimates that for every $20 increase in rent, one of the units will become vacant. The owner sets aside $60 per month from each of the occupied units to establish a repair fund. What rent should be charged per month in order to maximize the owner's profit if there are no other expenses? What is the owner's maximum monthly profit? How many units are occupied?
4. If it costs $1000 to manufacture 200 gizmos, then the average cost to manufacture each gizmo is 
or $5. Assume that the cost to manufacture x gizmos is given by the function 
. Find the number of gizmos that should be manufactured to minimize the average cost per gizmo.
5. After some waste has been dumped into a pond, the oxygen level in the water initially decreases but as the waste is oxidized, the amount of oxygen returns close to its former level over time. Assume that the normal amount of oxygen in the pond is considered to be 1, and that the oxygen level x days after waste is dumped into the pond can be found by the function 
, 
. After how many days is the oxygen level at its lowest level? When is the oxygen level at its highest level?
Suggestions/Extensions
If you do not have the Journey Through Calculus software, conduct an interactive introduction with the students by using a graphing calculator. When the expression for the time taken is found to be 
, enter the function in the Y= portion of the calculator.
Then press 2nd WINDOW to set up the table so that you can input your x values one at a time as you choose – see below.
Now you can begin to guess x values that might minimize the lifeguard's times. Some sample guesses are shown below.
Remember that these x values represent the distance that the lifeguard does not run – see the diagram on the previous page.
In exploring this problem, the students should be encouraged to model/represent the problem in different ways to increase their understanding of the problem.
Try problem 7 on this page if the rectangle is inscribed in the right triangle as shown.
Initiate a class project around the pop can problem. Have the students choose a can containing a favourite food item for which the diameter is clearly not the same as the height. Have the students write a letter to the manufacturer of that product to inquire as to why the company does not conserve metal by using the ideal 
ratio (COM). Examine each student's letter for correct mathematical understanding as well as proper English usage. Mail only one letter to each company. You will be surprised with the replies – from coupons to actual mathematical discussions about how the chosen can shapes conserve energy during the canning process.
For research (IL): Why do bees build their honeycomb in the shape they do? What are they minimizing or maximizing?
How are wind tunnels used to aid in the design of a car?
To think about: Geese fly in a wedge or V formation. Is the angle of the wedge different each time they fly, or is there an optimal angle that reduces wind resistance?
Students may wish to work in pairs or small groups when starting optimization problems. Such groupings allow for the developing and screening of ideas and strategies. (PSVS, CCT)
Students can also research their areas of interest for common optimization situations and problems.
Objective
F.3 Back to Top
To determine the instantaneous velocity and acceleration of a particle given the function for its position.
Instructional Notes
Having dealt with rates of change in F.1, students should be able to make the transition to finding the instantaneous velocity and acceleration with little difficulty.
In this section we are dealing with motion that is along a straight line and not in two or three dimensions. Consider a particle's position along this straight line relative to a fixed reference point, the origin of our measurements. It is common for this straight line to be thought of as horizontal (like the horizontal number line) or vertical (like the vertical number line). If the particle is to the right of (or above) the origin, then its position is considered positive, while if the particle is to the left of (or below) the origin, its position is considered to be negative.
To establish that the velocity and acceleration functions are the first and second derivatives of the position function, use an example such as the one that follows. Challenge the students a day or two before this class lesson to see if they could find the instantaneous velocity at 
.
Instructional Example: A very “hot” car moves in a straight line so that its position after t seconds is given by the function 
.
(a) Give the students the time values and have them complete the position portion of the table.
(b) Next have students graph the position as a function of time.
(c) Ask them to determine the average velocity between 
and 
and to interpret the graphical significance of that result.
Define: 
. Thus, 
. This is the slope of the secant line connecting the points 
and 
.
(d) Determine the instantaneous velocity when 
. The average velocity between 
and 
is given by 
. To find the instantaneous velocity at 
, take the average velocity over shorter and shorter time intervals. To do so, find 
or 
, the derivative of 
when 
. Since 
, 
m/s. This is the slope of the tangent line to the position-time graph at the point 
.
(e) Now that students know that the instantaneous velocity is given by 
, they can complete the velocity portion of the table below. Introduce the notation 
to describe the instantaneous velocity at time t .
Point out to the students that the average velocity between 
and 
was 49m/s and this is not the same as averaging 
and 
which gives 
. Average velocity is not an average of two velocities .
(f) Draw a graph of the velocity function 
.
(g) Define: 
. Ask the students to find the average acceleration between 
and 
and interpret the graphical significance of that result. 
or 
. This is the slope of the secant line through the two points 
and 
.
(h) Now find the instantaneous acceleration when 
. To do so, take the average acceleration over shorter and shorter time intervals. Thus 
. Since 
, 
. Thus 
. Note that this is the slope of the tangent line to the velocity-time graph at the point 
. Introduce the symbol 
for instantaneous acceleration.
(i) Finally, complete the instantaneous acceleration portion of the table.
Conclude by drawing the graph of the instantaneous acceleration function.
(j) Summarize: 
; 
, thus 
or 
and 
.
Examples/Activities
Have the students solve a few typical x -axis motion problems to develop confidence. Be careful of the function you create for the position. If the function is not always increasing for 
, then the particle will “double back” or “double-double back” creating some severe headaches. Save that type of function for the students that need to be challenged.
Sometimes students think that “after 4 seconds” is equivalent to 
. Be prepared to have the discussion about the start of time being 
not 
. It is obvious, but contradicts many of the students' past experiences (e.g., we have day 1 not day 0, 12:00 goes to 1:00 not 0:00).
1. Determine the velocity and acceleration functions for the given position functions.
(a)
(b)
(c)
2. A particle moves along the x -axis so that its position in metres after t seconds is given by the function 
. Find:
(a) The velocity and acceleration at any time t .
(b) The velocity when
.
(c) The acceleration when
.
(d) The position of the particle when the velocity is 66 m/s.
(e) The velocity of the particle when the acceleration is
.
The next type of problem is more practical.
3. A stone is shot upwards with a slingshot from the top of a hotel so that the stone's height above ground in metres t seconds after being released is given by the function 
.
(a) How high above the ground is the stone initially?
(b) When does the stone reach its maximum height?
(c) What is the maximum height reached by the stone?
(d) When does the stone hit the ground?
(e) With what velocity does the stone hit the ground?
(f) Explain the negative sign on the answer in part (e).
(g) What is the acceleration experienced by the stone?
4. If a projectile is fired vertically upwards from ground level, its height is given by the function 
where 
is the initial velocity of the projectile in m/s and g is the acceleration of gravity.
(a) Find the maximum height reached by a projectile on the earth if its initial velocity is 490 m/s. Use
.
(b) Find the maximum height reached by the projectile in part (a) on the moon if the moon's acceleration due to gravity is one-fifth that of the earth's.
5. Using the function 
, determine the initial velocity required for a projectile to reach a maximum height of 1960 m if launched from the earth's surface. Use 
.
6. The muzzle velocity of a shell is 375 m/s (initial velocity or velocity with which the shell leaves the barrel of the gun). If the shell is fired vertically upwards, and if the fuselage of a plane can withstand bullets travelling at a speed of no more than 32 m/s, what is the minimum height a plane can safely fly if being shelled? Use 
with 
.
Try the following distance, velocity, and acceleration calculus lab project.
Materials needed: A bicycle, several timing devices (wristwatch timers or stopwatches) as many pylons as timing devices, a measuring tape (100 m if possible).
Location: A hill nearby the school – is there a man-made hill on your schoolyard?
What happens: Starting at the top of the hill, place pylons a uniform distance apart (every 10 m? depending on how many you have) down the slope of the hill. A student with a timing device is to be located at each pylon. At a given signal, a student begins coasting down the hill from an at rest position. As the bicycle passes each pylon, the student at each pylon records the time.
Back in the Classroom: The time-distance data that has been collected is entered into the lists of the graphing calculator, and a scatter plot of the data is drawn. Fit a curve to the data – preferably cubic or quartic, but whatever fits best. Knowing the equation of position versus time, determine the velocity and acceleration functions. You can graph these and determine when the velocity was increasing or decreasing. Similarly, determine when the acceleration was positive or negative. Then, create a series of questions asking the students to determine how far the bicycle coasted before it reached a certain speed, what was the acceleration when the speed was a given value, and other questions you might create.
A similar experiment could be conducted at the curling rink. Students could take distance-time measurements of the rock from when it crosses the first hog line until it comes to rest – assuming a draw shot.
Suggestions/Extensions
For the more capable student, explore when an object is moving towards the origin, when an object is moving away from the origin, when the speed is increasing, when the speed is decreasing.
Some resources will show that the object is moving away from the origin when 
. (the position is positive and the velocity is positive or the position is negative and the velocity is negative).
The object is moving towards the origin if 
.
The object is speeding up if 
(the velocity is positive and the acceleration is positive or the velocity is negative and the acceleration is negative; that is, the acceleration is acting in the same direction as the velocity).
The object is slowing down if 
(the acceleration is acting in opposition to the velocity).
If a particle's position along the x -axis is given by the function 
for 
, the particle moves away from the origin for 
while it moves towards the origin for 
. It is speeding up for 
and slowing down for 
.
Have students create their own variation of the bicycle or curling rock experiment based on activities they are engaged in.
See Appendix A for additional objectives and activities.
Objectives
F.4 Back to Top
To solve related rate problems.
To represent problems and understandings through a variety of communication modes (COM).
To find alternate solutions and interpretations (CCT).
To engage in activities that require exploration and manipulation in order to develop understandings related to rate of change (CCT).
Instructional Notes
To help students achieve this objective you should have done implicit differentiation in D.10. If you choose to focus on this objective, it will take three to four class periods to complete.
Of the many applications of calculus, this one is perhaps the most frustrating for students. It is probably wise to first tackle problems dealing with two-dimensional figures such as squares, rectangles, and triangles. Then, introduce the students to problems involving spheres. Next, move to cylinders, rectangular boxes, and cones. Conclude with right triangle related rate problems.
In complicated problems, students will experience more success if they draw both a “whenever” diagram as well as a “when” diagram (COM). In the whenever diagram, students should use variables to label all parts of the diagram that are changing. From this diagram, a relationship needs to be found between/among these variables (CCT). After differentiating this relationship implicitly relative to time, students should draw a when diagram showing the value of each variable at the “when” moment of the problem. It is often necessary to take a side-trip in order to find the value of one of the variables. After these values are substituted into the differentiated expression, the quantity to be determined is found.
Here are some demonstration ideas that you or the students could conduct to give them a feel for what this objective is all about (CCT).
• Fill spherical balloons with water from a tap flowing at a uniform rate, watching to see if the radius grows more quickly early in the process or later on.
• From a tap flowing at a uniform rate, fill a clear conical container (funnel from the chemistry lab) with water and observe to see if the water in the cone rises more quickly early in the process or later on.
• Repeat the above experiment letting water flow into a clear cylindrical jar or glass.
• Turn over an hourglass timer to observe how the height of the sand changes as the sand flows from one side to the other.
• Place an eight foot 2 by 4 board vertically against the wall. Attach a cord to the base of the 2 by 4 and pull the base away from the wall at a uniform rate. Does the top of the board descend more quickly initially or later when the board is almost flat on the floor?
• Have two students walk on paths that are at right angles to one another, each at a uniform rate but not so as to collide. The rest of the class is to observe how the distance between the walkers is changing if both walkers head towards the intersection; one heads towards the intersection, the other heads away from the intersection. If both walkers hold onto the ends of a retractable dog-leash, then the stretching of the leash will help to visualize how the distance between them is changing.
The expected detail and format of a related rate problem is shown through a few examples.
Example 1: Oil is leaking from a sunken tanker and is forming a circular ring whose radius is increasing at a rate of 10 m/minute. How are the circumference of the ring and the area of the ring changing when the radius of the ring is 600 m?
Solution:
Draw a sketch of the “whenever” situation showing all variables.
Find an equation that relates the variables.
Differentiate both sides implicitly with respect to time “ t ”. Note that to measure a rate of change, you need to always involve time. Putting in the 1's is not necessary, but it does help some students to see that differentiation has occurred.
After differentiation, substitute in the “when moment” information. Include the units – they will show if things are on track.
Note that the rate of change in the circumference is independent of the size of the radius.
Conclude with a sentence:
• The circumference is increasing at a rate of 
m/min.
• The area is increasing at a rate of 
.
Example 2: The length of a rectangle is increasing at a rate of 6 cm/s. If the area of the rectangle is not changing, at what rate is the width of the rectangle decreasing when the length is 14 cm and the width is 7 cm?
Solution: Draw a sketch of the “whenever” situation showing all variables.
Find an equation that relates the variables.
Differentiate both sides implicitly with respect to time “ t ”. You will need the product rule.
Substitute in the “when moment” information. Note that if the area is not changing, its rate of change is 0.
Solve for 
.
Conclude with a sentence: The width is decreasing at a rate of 
.
Example 3: A spherical balloon is filled with 
of helium. A leak in the balloon causes the helium to escape at a rate of 
. At what rate was the radius of the balloon decreasing 49 minutes after the leak began?
Solution: Draw a sketch of the “whenever” situation showing all variables.
Find an equation that relates the variables.
Differentiate both sides implicitly with respect to time “ t ”.
.
Substitute in the “when moment” information. Note that we do not know the radius at the “when moment”. We will have to go on a side trip to find the radius.
SIDE TRIP to find the radius: If the balloon held 4500 
and lost 
for 49 minutes, then the amount of helium remaining in the balloon is 
. The remaining helium is still in a spherical balloon whose volume is 
. Thus, 
. Solving for r , 
. Thus, 
.
Return to and substitute in the value for r and solve for 
.
Conclude with a sentence: The radius is shrinking at a rate of 
.
Example 4: Water is being poured into a conical vase at a rate of 
. The diameter of the cone is 30 cm and the height of the cone is 25 cm. At what rate is the water level rising when the water's depth is 20 cm?
Solution: Draw a sketch of the “whenever” situation showing all variables.
Find an equation that relates the variables.
Use similar triangles to find a relationship between r and h . Solve for r and substitute into . The question asks us to find the rate at which the water level is rising, so we try to obtain an expression for the volume in terms of only the height of the water.
Differentiate both sides implicitly with respect to time “ t ”.
Substitute in the “when moment” information.
Solve for 
.
Conclude with a sentence: The water level is rising at a rate of 
when the depth of the water is 20 cm.
Example 5: A car, travelling north at 48 km/h is approaching an intersection. A truck, travelling east at 60 km/h is moving away from the same intersection. How is the distance between the car and the truck changing when the car is 9 m from the intersection and the truck is 40 m from the intersection?
Solution: Draw a sketch of the “whenever” situation showing all variables (COM).
Find an equation that relates the variables.
Differentiate both sides implicitly with respect to time “ t ”.
Substitute in the “when moment” information. Note that 
is negative because y is decreasing as the car approaches the intersection.
Note that we must find the value of z at the “when moment”. Draw a “when” diagram and use the theorem of Pythagoras.
Substitute 
into and solve for 
.
Conclude with a sentence: The distance between the car and truck is increasing at a rate of 48 km/h.
These five examples are not a complete sampling of all the different types of related rate problems. Some of the more common types of problems missing from the list are the intersection problems in which one of the sides in the right triangle is fixed, while the other two change. Also missing from the list are problems dealing with the filling/draining of three-dimensional objects whose sides are uniform (cylinders and rectangular boxes).
Suggestions/Extensions
Have the students create actual models and carry out experiments to better relate to the problem situations (CCT).
Also, when students obtain an answer such as a rate of 
, challenge the students to make sense of the answer (and not just as a decimal number, although that may be the starting point).
