2.6 Related Rates

Don’t get

Ex. Two rates that are related.

Related rate problems are differentiated withrespect to time. So, every variable, except t is differentiated implicitly.

Given y = x2 + 3, find dy/dt when x = 1, giventhat dx/dt = 2.

y = x2 + 3

dt

dxx

dt

dy2= Now, when x = 1 and dx/dt = 2, we

have

4)2)(1(2 ==dtdy

Procedure For SolvingRelated Rate Problems

1. Assign symbols to all given quantities and quantities to be determined. Make a sketch

and label the quantities if feasible.

2. Write an equation involving the variableswhose rates of change either are given or areto be determined.

3. Using the Chain Rule, implicitly differentiateboth sides of the equation with respect to t.

4. Substitute into the resulting equation all knownvalues for the variables and their rates of change.Solve for the required rate of change.

Ex. A pebble is dropped into a calm pond, causingripples in the form of concentric circles. The radiusr of the outer ripple is increasing at a constant rateof 1 foot per second. When this radius is 4 ft., whatrate is the total area A of the disturbed water increasing.

Given equation: 2rA π=

Givens: 41 == rwhendtdr

Differentiate:

dt

drr

dt

dA π2= ( )( )412π=dt

dA

π8=

?=dtdA

An inflating balloon

Air is being pumped into a spherical balloon at therate of 4.5 in3 per second. Find the rate of changeof the radius when the radius is 2 inches.

Given:sec/5.4 3in

dt

dV= r = 2 in. ?: =

dtdr

Find

Equation: 3

3

4rV π=

Diff.& Solve: dt

drr

dt

dV 24π=dt

dr2245.4 π=

€

.09in /sec =dr

dt

The velocity of an airplane tracked by radar

An airplane is flying at an elevation of 6 miles on a flightpath that will take it directly over a radar tracking station. Let s represent the distance (in miles)between the radarstation and the plane. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the velocity of the plane.

s

x

6

Given:

Find:

Equation:

Solve:

10400 =−= sdtds

?=dtdx

x2 + 62 = s2

dt

dss

dt

dxx 22 =

To find dx/dt, wemust first find xwhen s = 10

836100362 =−=−= sx

( ) ( )( )40010282 −=dt

dxmph

dt

dx500−=

Day 1

A fish is reeled in at a rate of 1 foot per secondfrom a bridge 15 ft. above the water. At what rate is the angle between the line and the water changing when there is 25 ft. of line out?

15 ft.

x

θ

Given:

Find:

Equation:

Solve:

1−=dtdx

x = 25 ft. h = 15 ft.

?=dtdθ

x

15sin =θ 115sin −= xθ

( )dt

dxx

dt

d 215cos −−=θ

θ

dt

dx

xdt

d

θθ

cos15

2

−=

( )1

2520

25

15

2

−

⎟⎠⎞

⎜⎝⎛

−=

dt

dθ

sec/100

3rad

dt

d=

θ

Ex. A pebble is dropped into a calm pond, causingripples in the form of concentric circles. The radiusr of the outer ripple in increasing at a constant rateof 1 foot per second. When this radius is 4 ft., whatrate is the total area A of the disturbed water increasing.

An inflating balloon

Air is being pumped into a spherical balloon at therate of 4.5 in3 per minute. Find the rate of changeof the radius when the radius is 2 inches.