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Related Rates Problems Note
Often it is required that we find the rate at which one variable is changing, given the rate of change of a related
variable.
Steps for Solving a Related Rate Problem
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Related Rate Problem involving a circular model
Eg.1: A pebble is thrown into a lake and causes a circular ripple to spread outward at a rate of 2 m/s. Find the
rate of change of the area in terms of π.
a) 3 seconds after the pebble is thrown.
b) When the area of the ripple is 9π m2
Solution:
1. Make a sketch.
2. Let Statements:
3.
Required to Find:
4.
Equation that relates the variables.
5. Implicitly differentiate both sides w.r.t. time.
6. Substitute in known quantities and known rates.
7. Solve the equation:
8. Concluding statement.
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 A  r 2 and A  9
b)
 r  ______ m
Related Rates Problems Note
Therefore the area of the circle is
Related Rate Problem involving a Right Triangle Model
Eg.2: A dog runs across a path at a rate of 4 m/s. A culvert is located directly beneath the dog. A fish swims
right underneath the dog at a rate of 3 m/s. in a direction perpendicular to the path of the dog. Find the rate of
change of the distance between the two animals 1 second later.
Solution:
1. Sketch
2. Let x represent the distance traveled by the fish (m)
Let y represent the distance traveled by the dog (m)
Let r represent the distance between the dog and the fish (m)
3. Required to find:
4. Equation:
5. Implicitly Differentiate w.r.t. time:
6. Substitute in known quantities and known rates.
7. Solve the equation:
8. Concluding Statement:
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Related Rates Problems Note
Related Rate Problem involving a Conical Model
Eg.3: Water is pouring into an inverted right circular cone at a rate of π m3/min. The height and the diameter
of the base of the cone are both 10 m. How fast is the water level rising when the depth of the water is 8 m?
Solution:
1. Sketch:
2. Let V represent the volume of water in the cone at time, t.
Let r represent the radius of water in the cone at time, t.
Let h represent the height of water in the cone at time, t.
3. Required to find:
4. Equation:
5. Before we differentiate implicitly we need to have our equation in terms of only h. Using similar
triangles from our sketch;
6. Substitute in known quantities and known rates.
7. Solve the equation:
8. Concluding Statement:
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Related Rates Problems Note
Related Rate Problem involving Similar Triangle Model
Eg.4: A student who is 1.6 m tall walks directly away from a lamppost at a rate of 1.2 m/s. A light is situated
8 m above the ground on the lamppost. Show that the student’s shadow is lengthening at a rate of 0.3
m/s. when she is 20 m from the base of the lamppost.
Solution:
1. Sketch:
2. Let x represent the length of the shadow.
Let y represent the distance the student is from the base of the lamppost.
3. We are given:
and are required to find:
4. To determine an equation that relates x and y, we use similar triangles.
5. Implicitly Differentiate both sides w.r.t. time:
6. Substitute in known quantities.
7. Solve the equation:
8. Concluding Statement:
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Related Rates Problems Note
Related Rate Problem involving a Rectangular Prism Model
Eg.5: The base of a rectangular tank is 3 m by 4 m and is 10 m high. Water is added at a rate of 8 m3/min.
Find the rate of change of water level when the water is 5 m deep.
Solution:
1. Sketch a rectangular prism and label.
2. Let V represent the volume of the tank.
Let l represent the length of the tank.
Let w represent the width of the tank.
Let h represent the height of the tank.
3. We are given the length, width and height of the tank and
Required to find
when the water is 5 m deep.
4. Equation that relates the variables.
Length and Width remain constant.
5. Implicitly Differentiate both sides w.r.t. time.
6. Substitute in known quantities.
7. Solve the equation:
8. Therefore the rate of change of water in the tank is
Homework: Related Rates Worksheet
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2
m/min.
3