Related Rates

Related rates problems involve finding arate at which a quantity changes by

relatingthat quantity to other quantities whoserates of change are known.

Because these problems involve rates, theymust be differentiated with respect totime.

What are related rates?

Quick Review on Implicit Differentiation

45: 223 xyyyExample

02523 2 xdxdy

dxdyy

dxdyy

xyydtdy 2)523( 2

)523(2

2

yyx

dxdy

We will determine how to solve problems involving:

I. CirclesII.SpheresIII.TrianglesIV.ConesV. Cylinders

Types of Problems

General ProcedureThe most common way to approach related rates problems is the following:

1. Identify the known variables, including rates of change and the rate of change that is to be found. **Drawing a picture or representation of the problem can help to keep everything in order**

2. Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found.

3. Differentiate both sides of the equation with respect to time. (Often, the chain rule is employed at this step.)

4. Substitute the known rates of change and the known quantities into the equation. Solve for the wanted rate of change.

**Errors in this procedure are often caused by plugging in the known values for the variables before (rather than after) finding the derivative with respect to time. Doing so will yield an incorrect result.**

Example 1Olympic swimmer Mr. Spitz leaps into a pool, causing ripples in the form of concentric circles. The radius is increasing at a constant of 4 feet per second. When the radius is 8 feet, at what rate is the total area of disturbed water changing (assuming the splash is negligible)?

The variables r and A are related by A=πr2. The rate of change of the radius r is dr/dt=4

Equation: A=πr2 Given rate: dr/dt=4 Find: dA/dt when r=8

][][ 2rdtdA

dtd

dtdrr

dtdA 2

)4)(8(2dtdA

sec642ft

dtdA

EXAMPLE 2Mr. Spitz, the circus clown, is traveling with the Barnum and Bailey circus, is told to inflate a balloon. If the volume increases at a constant rate of 50 cm3/sec, at what rate is the radius increasing when the volume is 972π cm3?Equation: V= 4/3πr3

Given: dV/dt=50 V = 972π Find dr/dt

3

34 rV

3

34972 r

3729 r

cmr 9

dtdrr

dtdV 24

dtdr2)9(450

sec0491.16225 cm

dtdr

Example 3aMajor Spitz is flying a rescue mission to extract General Earl during a battle in WWI. He flies his plane at 200 mph at an altitude of 6 miles. What is the rate of change of the shortest distance between them when the horizontal distance is 8 miles?

8 mi

6 midz/dt = ??

dx/dt = -200

222 yxz 22 )6()8( z

miz 10100

222 yxz

dtdxx

dtdzz 22

)200)(8(2)10(2 dtdz

mphdtdz 160

)10(2)200)(8(2

Example 3bGeneral Earl is watching for his rescue plane with binoculars. Major Spitz slows down to 100 mph and descends to 3 miles to spot General Earl. What is the rate of change of the angle the General is watching from when the horizontal distance is 4 miles?

xxy 3tan

dtdxx

dtd 22 3)(sec

dtdx

xdtd

2

2 )(cos3

)100)(4

)54(3

( 2

2

dtd

hrrad

dtd 12

25300

Example 4Mr. Spitz was best friends with Augustus Caesar. While studying by candlelight, Spitz was testing his speed with implicit differentiation and timed himself with an hourglass, the most advanced technology at the time. He knows that the sand initially falls forming a cone, whose radius is twice its height. The hourglass fills at a constant rate of 3 cm3/min. When the volume is 36π cm3, what is the rate of change of the height?

hrV 2

31

hhV 2)2(31

)4(31 3hV

dtdhh

dtdV ))(12(3

1 2

dtdh2)3(43

sec121

363 cm

dtdh

Note: When V = 36π, h = 3

Example 5Mr. Spitz is filling his cylindrical water jug with some “good H2O” before he runs the iron man triathlon. The radius is 4 inches; as Mr. Spitz fills the jug, the height is changing at a rate of .5 in/sec. What rate is the volume changing at?

hrV 2

hV 2)4(

dtdh

dtdV 16

sec8)5(.163in

dtdV

That’s All Folks!