Related Rates

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Related Rates

• Finding Related Rates ? Problem Solving with
  Related Rates

2
Related Rates Solving Differential Equations

• Find the indicated values for dy/dt and dx/dt.

3

• 4. The radius of a circle is increasing at a
  rate of 3 inches per second. Find the rate of
  change of the area of the circle when the radius
  is 3 feet.

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Guidelines for Solving Related-Rate Problems

• Identify all given quantities and quantities to
  be determined. Make a sketch and label the
  quantities.
• Write an equation involving the variables whose
  rates of change either are given or are to be
  determined
• Using the Chain Rule, implicitly differentiate
  both sides of the equation with respect to time
  t.
• After completing Step 3, substitute into the
  resulting equation all known values for the
  variables and their rates of change. Then solve
  for the required rate of change.

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Filling a Spherical Balloon

• A spherical balloon is inflated with gas at the
• rate of 20 ft3/min. Find how fast is the radius
  of
• the balloon increasing at the instant the radius
• is
• a) 1 ft b) 2 ft

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Organize, Identify, and Write an Equation

7
Gravel is falling onto a conical pile at a rate
of 10 cubic feet per minute. The diameter of the
pile is five times the altitude. Find the rate of
change of the height of the pile when the pile is
10 feet high.
8
The base of a 25-foot ladder is being pulled away
from the house it leans on at a rate of 4 feet
per second. At what rate is the top of the
ladder moving when the base of the ladder is 7
feet from the building?
25
y
x
25
24
7
9
Filling a Conical Tank

• A water tank has the shape of an
• inverted cone with base radius of
• 2 m and height of 4 m. If water is
• being pumped into the tank at a
• rate of 2 m3/min, find the rate at
• which the water level is rising
• when the water is 3 m deep.

2 m
4 m
3m
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Organize, Identify, and Write an Equation

• Since 2r h, and r h/2, substitute h/2 for r
  in order to
• have an equation in just V and h.

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1. Mr. Aldridge, who is 6 feet tall, walks a
rate of 4 feet per second away from a light that
is 13 feet above the ground. Find the rate at
which his shadows length is changing when he is
10 feet from the base of the light.
13 ft
6
x
s
d
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2. Mr. Aldridge, who is 6 feet tall, walks a
rate of 4 feet per second away from a light that
is 13 feet above the ground. Find the rate at
which the position of the tip of his shadow is
changing when he is 10 feet from the base of the
light.
13 ft
6
s
x
d