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AP Calculus BC
Unit 4: Contextual Applications of Differentiation
4.5 Solving Related Rates Problems
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What are related rates problems in calculus concerned with?
Rate of change over time
Related rates problems often involve finding the rate of change of one variable given the rate of change of
another
The equation for the area of a circle is
A
=
A =
A
=
π
r
2
\pi r^{2}
π
r
2
.
What is the derivative of the area of a circle
A
A
A
with respect to time
t
t
t
?
d
A
d
t
=
\frac{dA}{dt} =
d
t
d
A
=
2
π
r
d
r
d
t
2 \pi r \frac{dr}{dt}
2
π
r
d
t
d
r
If the radius
r
r
r
of a circle is increasing at 2 cm/s, then \frac{dr}{dt} = 2
Related rates problems require identifying
variables
and their relationships.
Steps to identify variables and relationships in related rates problems
1️⃣ Define each variable and its rate of change
2️⃣ Write down the equation relating the variables
What is the equation relating the area
A
A
A
and radius
r
r
r
of a circle?
A
=
A =
A
=
π
r
2
\pi r^{2}
π
r
2
The equation
A
=
A =
A
=
π
r
2
\pi r^{2}
π
r
2
relates the area
A
A
A
of a circle to its radius
Drawing a diagram in related rates problems helps visualize the relationships between
variables
.
What is one key benefit of drawing a diagram in related rates problems?
Clarifies the problem
Match the geometric property with the corresponding variable in a cone-shaped tank
Height ↔️ h
Radius ↔️ r
Volume ↔️ V
Stating the given information in related rates problems ensures clarity and
accuracy
What is an example of given information in a related rates problem?
d
r
d
t
=
\frac{dr}{dt} =
d
t
d
r
=
2
2
2
If
d
r
d
t
=
\frac{dr}{dt} =
d
t
d
r
=
2
2
2
cm/s, then \frac{dA}{dt} = 4 \pi r</latex> cm²/s.
Why is drawing a diagram useful in related rates problems?
Visualizes the relationships
What is the equation relating the area
A
A
A
and the radius
r
r
r
of a circle?
A
=
A =
A
=
π
r
2
\pi r^{2}
π
r
2
The area of a circle is given by the equation
A
Diagrams can help clarify the context of
related rates
problems.
Steps to use diagrams in related rates problems
1️⃣ Visualize the problem
2️⃣ Label variables and rates of change
3️⃣ Clarify relationships between variables
What is the formula for the volume of a cone in terms of its radius
r
r
r
and height
h
h
h
?
V
=
V =
V
=
1
3
π
r
2
h
\frac{1}{3} \pi r^{2} h
3
1
π
r
2
h
Stating the given information in related rates problems is essential for
accuracy
If the radius of a circle is
r
=
r =
r
=
5
5
5
cm and its rate of change is
d
r
d
t
=
\frac{dr}{dt} =
d
t
d
r
=
2
2
2
cm/s, what information has been stated?
Radius and rate of change
In related rates
problems
, stating what needs to be found provides a key objective for solving the problem.
Clearly stating what needs to be found in related rates problems is crucial for defining the
objective
What is the area
A
A
A
of a rectangle where the length
l
l
l
is twice the width
w
w
w
?
A
=
A =
A
=
l
2
2
\frac{l^{2}}{2}
2
l
2
Establishing the equation relating the variables is a critical step in solving
related rates
problems.
To establish the equation in related rates problems, the first step is to identify known
quantities
What are related rates problems in calculus designed to find?
Rate of change of one variable
In related rates problems, if
d
r
d
t
=
\frac{dr}{dt} =
d
t
d
r
=
2
2
2
cm/s, then \frac{dA}{dt} = 4\pi r</latex> cm²/s. This result is obtained by differentiation
Steps to solve related rates problems
1️⃣ Define variables and rates of change
2️⃣ Write the equation relating variables
3️⃣ Differentiate the equation with respect to time
4️⃣ Substitute known values
5️⃣ Solve for the target rate of change
Stating the given information in
related rates problems
ensures clarity and accuracy.
What are the values of
r
r
r
and
d
r
d
t
\frac{dr}{dt}
d
t
d
r
in the example of a circle with a changing radius?
r
=
r =
r
=
5
5
5
cm,
d
r
d
t
=
\frac{dr}{dt} =
d
t
d
r
=
2
2
2
cm/s
In related rates problems, the target rate of change is written using differential
notation
The first step in stating given information is to identify the
variables
What should numerical values in related rates problems always include?
Units
Steps to state given information in related rates problems
1️⃣ Identify variables
2️⃣ Write down their values
3️⃣ Use a table
Clearly stating what needs to be found is crucial for defining the objective of
related rates problems
.
The target rate of change is written using differential
notation
What should the target rate of change always include?
Units
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