Math 100 - Related Rates - *not to hand in*
Basic skills required to work through the problems:
• differentiate (explicitly or implicitly) a mathematical expression;
• apply Pythagoras’ theorem;
• write down proportionality relations between the sides of similar triangles;
• use volume formulas to find the volume of simple solids.
Learning Goals: After completing this problem set, you should be able to apply the
problem-solving strategy below and solve problems involving related rates.
Problem-Solving Strategy: Steps to approach a problem.
1. Understand the problem: Read the problem and make sure you understand
it clearly. Ask yourself:
• What do I know?
– What are the given quantities?
– What are the given conditions?
• What do I need to find?
–
–
–
–
What is the problem asking?
What is the unknown?
Do I need to introduce suitable notation?
Do I need to draw a sketch?
2. Think of a plan: Find the connection between the given information and the
unknown. Ask yourself:
• What calculus concept(s) do I need to apply?
– Do I need to differentiate? With respect to what variable do I differentiate?
– Do I need to solve an equation? Why? What variable do I solve for?
– Do I need to apply a certain theorem or definition? Why?
3. Carry out your plan: Do your computations!
4. Reflect on what you are doing: Does your answer make sense?
1. (WITH SOLUTION) Read the following problem and answer the questions below.
Remember to draw a diagram!
“Two roads, one in the east-west direction and the other north-south, intersect
at right angles. Car A is 100 km directly south of the intersection point of the
roads and is travelling north at 80 km/h. Car B is 60 km west of the intersection
and is travelling west at 50 km/hr. At what rate is the distance between the cars
changing? You do not need to simplify your answer in this question.”
(a) UNDERSTAND THE PROBLEM: What is the problem asking? Choose the
correct answer from the ones below.
(a.1) The problem asks to find the rate of change of the distance of Car A from
the intersection point at a certain moment in time.
(a.2) The problem asks to find the rate of change of the distance of Car B from
the intersection point at a certain moment in time.
(a.3) The problem asks to find the rate of change of the distance of Car A from
Car B at a certain moment in time.
(a.4) The problem asks to find the rate of change of the distance between the
cars at any time.
Solution: The correct answer is (a.3). The problem asks to find the rate
of change of the distance of Car A from Car B at a certain moment in
time. We have information about the cars at a specific moment of time
when Car A is 100 kilometres directly south of the intersection and driving
north at 80 km/hr and Car B is 60 kilometres west of the intersection and
driving west at 50 km/hr, so any calculations involving this information
will be referring to that specific moment of time.
(b) UNDERSTAND THE PROBLEM: Assign names to variables. For example,
suppose D is the distance between the cars, x is the distance of Car A from the
intersection, y is the distance of Car B from the intersection. Draw a picture
if necessary. As you determined above, you need to find the rate of change of
= ..., dy
= .... What are the
D at a specific time when x = ..., y = ..., dx
dt
dt
missing values in this sentence?
Solution: Placing the origin of the axes in the intersection point and
taking the positive x-axis to point east of the intersection and the positive
y-axis to point north of the intersection, then the problem asks for the rate
= +50, and
of change of D at a specific time when x = 60, y = 100, dx
dt
dy
= −80 (negative because the distance from car B to the intersection is
dt
decreasing).
(c) THINK OF A PLAN: Which of the following actions best describes a strategy
to solve the given problem? Choose the correct answer from the ones below.
(c.1) Finding the derivative of D with respect to x.
(c.2) Finding the derivative of D with respect to y.
(c.3) Finding the derivative of D with respect to time.
Solution: The correct answer is
(c.3) Finding the derivative of D with respect to time.
(d) THINK OF A PLAN: Find an expression relating D to x and y.
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Solution: If you draw a picture of the position of the cars relative to the
intersection, you find that D2 = x2 + y 2 . This relation is true at any
time (assuming the cars continue to drive in the same direction). Since
both cars are moving, all the distances x, y, and D are functions of time.
Thus the expression found in the previous part can be written as [D(t)]2 =
[x(t)]2 + [y(t)]2 .
(e) THINK OF A PLAN: Which of the distances defined in part (b) changes with
time?
(f) CARRY OUT YOUR PLAN: Use all the information collected in parts (a) (e) and solve the given problem.
(g) REFLECT ON WHAT YOU ARE DOING: What units is your answer expressed in? Make sure that your calculation is dimensionally consistent.
Solution: Based on the strategy determined above we need to find dD
.
dt
We do not have an explicit expression for D as a function of time so we
differentiate implicitly the equation found in part (c). So
dD
dx
dy
= 2x + 2y
dt
dt
dt
1
dy
dD
dx
=
2x + 2y
dt
2D
dt
dt
p
√
Using Pythagoras we find that D = x2 + y 2 = 10000 + 3600 so
2D
dD
1
√
(2 × 50 × (60) + 2 × 100 × (−80))
=
dt
2 × 13600
from which we conclude
dD
dt
=
−5000
√
13600
km/hr.
2. (Warm up) Suppose an ice sculpture has a conical shape. The sculpture was left in
the sun and is now slowly melting in such a way that the height of the ice cone is
always equal to its diameter. Let r be the radius of the cone at a certain time. Find
an expression for the volume of the cone V as a function of r. No other variables,
besides r, should appear in your answer.
3. A container whose shape is a right circular cone is turned upside-down (vertex is
at the bottom) and used to store sand; the height of cone is 16 m, the diameter of
the top is 6 m. The container is filled half-way (by depth) with sand draining out
at a constant rate of 3 m3 per second. How fast is the depth of the sand changing
at that time? Hint: Draw a diagram.
4. An extension ladder can lengthen or shorten. An extension ladder is leaning against
a wall. The ladder has its top end fixed on a hook 15m above the ground on the
vertical wall. The bottom of the ladder is on the ground, 8m from the base of the
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wall. The bottom of the ladder is slipping and moving at 2 m/min away from the
base of the wall. How fast is the ladder lengthening?
5. A circular ferris wheel with radius 10 metres is revolving at the rate of 10 radians
per minute. How fast is a passenger on the wheel rising when the passenger is 6
metres higher than the centre of the wheel and is rising? Include units with your
answer.
6. At a certain instant an aircraft is flying due east at 400 km/hr and passes directly
overhead a car traveling due southeast at 100 km/hr on a straight, level road. If
the aircraft is traveling at an altitude of 1 km, how fast is the distance between
the aircraft and the car increasing 36 seconds after the airplane passes directly over
the car? Hint: Draw a very careful diagram! You shall also need the Cosine Law:
For a triangle with sides a, b, and c and the angle µ opposite the side c, one has
c2 = a2 + b2 − 2ab cos(µ).
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