4.6 Related Rates
Objective:
SWBAT solve related rate problems
Related Rates
• When one or more values in an equation
change over time, we have related rates.
• We simply write equations that you might
have written prior this course, then
differentiate them with respect to time, t.
• It is so much fun, you are going to love the
process!!! That’s a Carucci guarantee.
Remember that this was all with respect to x.
In order to do this, we are going to have to be given
information to find values for all other variables in the
problem.
• Related rate problems typically fall into three
categories:
– Pythagorean Theorem
– Right-Triangle Trigonometry
– Know or given formulas involving Volume, Area, or
other scenarios
• Answers need to be written using the correct
units!!!
• On a free-response question on the AP exam,
units are usually worth 1 out of 9 points. This
means that if you get the entire problem
wrong but still include the correct units, you
will at least pick up one point!
Example 3: Tweety is resting in a bird house 24 feet off the
ground. Using a 26 foot ladder which he leaned against the
pole holding the bird house, Sylvester tries to steal the
small yellow bird. Tweety’s bodyguard, Hector the dog,
starts pulling the base of the ladder away from the pole at
a rate of 2 ft/s. How fast is the ladder falling when it is 10
feet off the ground?
We now must take the derivative with
respect to time.
We now have two rates: how
fast x is changing and how fast
y is changing.
We are still missing a value
for x. How can we find it?
Often times in these situations you
will need to substitute back into the
original equation.
Example 4: A police cruiser, approaching a right-angled intersection
from the north, is chasing a speeding car that has turned the corner
and is now moving straight east. When the cruiser is 0.6 miles north
of the intersection and the car is 0.8 miles to the east, the police
determine with radar that the distance between them and the car is
increasing at 20 mph. If the cruiser is moving at 60 mph at the instant
of measurement, what is the speed of the car?
Example 5: Bugs Bunny finished his final act, and then began dancing
off stage with a spotlight covering his every move. If he is moving off
the stage along a straight path at a speed of 4 ft/s, and the spotlight
is 20 ft away from the path, what rate is the spotlight rotating when
Bugs is 15 feet from the point on the path closest to the spotlight?