Related Rates

A fun and exciting application of derivatives

The Study of Change

• Used to work with real life problems where there is more than one variable such as– Rain pouring into a pool

• How fast is the height changing compared to the speed the volume is changing?

– Falling ladder• How fast is the base moving away from the house compared

to the speed the top of the ladder is falling towards the ground?

– Distance between two moving objects• How fast does the distance between the objects change

compared to the speed of each car?

The Ladder Problem

An 8 foot long ladder is leaning against a wall.  The top of the ladder is sliding down the wall at the rate of 2 feet per second.  How fast is the bottom of the ladder moving along the ground at the point in time when the bottom of the ladder is 4 feet from the wall ?

Animation(Hopefully)

• http://www2.scc-fl.edu/lvosbury/images/LadderNS.gif

Example

• Two cars travel on perpendicular roads towards the intersection of the roads. The first car starts 100 miles from the intersection and travels at a constant rate of 55 mph. The second car starts at the same time, 250 miles from the intersection and travels at a constant speed of 60 mph. How fast it the distance between them changing 1.5 hours later?

» From Teaching AP Calculus, McMullin

Two Different Solutions

• Let t = time traveled• X = 100 – 55t• Y = 250 -60t• Z(t) =

y

x

z

22 )60250()55100( tt

Differentiate 22 )60250()55100( tt

62.65

1605.172

)60)(160(2)55)(5.17(2

)60250()55100(2

)60)(60250(2)55)(55100(2

22

22

dt

dz

dt

dz

tt

tt

dt

dz

Method 2—Easier?

• Differentiate at start with Pythagorean Thm

222 yxz

22 1605.172

)60)(160(2)55)(5.17(22

22

222

dt

dzz

dtdyy

dtdxx

dt

dz

dt

dyy

dt

dxx

dt

dzz

Compare Un-Simplified Versions

22 1605.172

)60)(160(2)55)(5.17(22

22

222

dt

dzz

dtdyy

dtdxx

dt

dz

dt

dyy

dt

dxx

dt

dzz

62.65

1605.172

)60)(160(2)55)(5.17(2

)60250()55100(2

)60)(60250(2)55)(55100(2

22

22

dt

dz

dt

dz

tt

tt

dt

dz

What units?

• The distance between the two cars is changing at a rate of -65.62 miles per hour

• In general, units of the derivative

• units of f(x)/ units of independent variable

Simplified Example

• Suppose x and y are both differentiable functions of t and are related by the equation

• Find dy/dt when x =1, given that dx/dt =2 when x = 1

» From Calculus, 8th e, Larson

32 xy

Solution

• Use Implicit Differentiation

• When x = 1 and dx/dt =2,

dt

dxx

dt

dy2

4)2)(1(2 dt

dy