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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 xdx
dy
dx
dyy
dx
dyy
xyydt
dy2)523( 2
)523(
22
yy
x
dx
dy
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
][][ 2rdt
dA
dt
d
dt
drr
dt
dA 2
)4)(8(2dt
dA
sec642ft
dt
dA
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
3
4rV
3
3
4972 r
3729 r
cmr 9
dt
drr
dt
dV 24
dt
dr2)9(450
sec0491.162
25 cmdt
dr
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
dt
dxx
dt
dzz 22
)200)(8(2)10(2 dt
dz
mphdt
dz160
)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?
xx
y 3tan
dt
dxx
dt
d 22 3)(sec
dt
dx
xdt
d2
2 )(cos3
)100)(4
)54(3
(2
2
dt
d
hrrad
dt
d12
25
300
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
dt
dhh
dt
dV))(12(3
1 2
dt
dh2)3(43
sec12
1
36
3 cmdt
dh
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(
dt
dh
dt
dV 16
sec8)5(.163in
dt
dV
That’s All Folks!