Related Rates

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Related Rates

• 3.7

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Finding a rate of change that cannot be easily
measured by using another rate that can be is
called a Related Rate problem.
Steps for Related Rates Problems
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.
3
A ladder, 10 ft tall rests against a wall. If the
ladder is sliding away from the bottom of the
wall at 1 ft/sec, how fast is the top of the
ladder coming down the wall when the bottom is 6
ft from the wall?
We want dy/dt when x 6
10
y
x
The ladder is moving down the wall at ¾ ft/sec
when it is 6 ft. from the wall.
At x 6, y 8 by Pythagorean theorem
4
Water is draining from a cylindrical tank at 3
liters/second. How fast is the surface dropping?
(We need a formula to relate V and h. )
(r is a constant.)
5
Hot Air Balloon Problem
Given
How fast is the balloon rising?
Find
6
Hot Air Balloon Problem
Given
How fast is the balloon rising?
Find
7
Air is being pumped into a balloon at a rate of
100 cubic cm /sec. How fast is the radius of the
balloon increasing when the diameter is 50 cm?
We want dr/dt when d50 or r 25
8

• Batman and Scooby Doo are having lunch together
  when they both simultaneously receive a call.
  Batman heads off to Gotham city traveling east at
  40 miles per hour. Scooby hops in the mystery
  machine and heads north at 30 miles an hour. How
  fast is the distance between them changing 6
  minutes later?

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The batmobile travels east at 40 mi/hr. The
mystery machine travels north at 30 mi/hr.
How fast is the distance between the vehicles
changing 6 minutes later?
B
A
10
Batman travels east at 40 mi/hr. Scooby travels
north at 30 mi/hr.
How fast is the distance between the vehicles
changing 6 minutes later?
B
A