Section 4.6a

RELATED RATES

Do NowWrite the formula for distance from the origin to a point… and find its derivative!

2 2

12 2

2

dD dx dyx y

dt dt dtx y

2 2D x y

2 2

dx dyx ydt dtx y

Suppose that the coordinates of the point (x and y) are bothdifferentiable with respect to t. Then we can use the ChainRule to find an equation that relates dD/dt, dx/dt, and dy/dt:

Any equation involving two or more variables that aredifferentiable functions of time t can be used to find anequation that relates their corresponding rates!!!

Related Rates Problems

Questions like these require us to calculate arate that may be difficult to directly measure,using a rate that we know, or is easy tomeasure RELATED RATES!!!

Examples:• How fast is a balloon rising at a given instant?• How fast does the water level drop when a tankis drained at a certain rate?• How fast does the surface area of a bubbleincrease when its volume increases at a certainrate?

Related Rates Problem Strategy1. Draw a picture and name the variables and constants. Use t for time. Assume all variables are differentiable functions of t.

2. Write down the numerical information (in terms of the symbols you have chosen).

3. Write down what we are asked to find (usually a rate, expressed as a derivative).

Related Rates Problem Strategy4. Write an equation that relates the variables. You may have to combine two or more equations to get a single equation that relates the variable whose rate you want to the variables whose rates you know.

5. Differentiate with respect to t. Then express the rate you want in terms of the rate and variables whose values you know.

6. Evaluate. Use known values to find the unknown rate.

Guided PracticeWhen a circular plate of metal is heated in an oven, itsradius increases at the rate of 0.01 cm/sec. At what rateis the plate’s area increasing when the radius is 50 cm?

Step 1: r = radius of plate, A = area of plate

Step 2: At the instant in question, dr/dt = 0.01 cm/sec, r = 50 cm

Step 3: Need dA/dt

Step 4:2A r

Guided PracticeWhen a circular plate of metal is heated in an oven, itsradius increases at the rate of 0.01 cm/sec. At what rateis the plate’s area increasing when the radius is 50 cm?

Step 5: 2dA dr

rdt dt

Step 6: 2 50 0.01dA

dt

At the instant in question, the area is increasing at the rate of

2cm sec

2cm sec

Guided PracticeA hot-air balloon rising straight up from a level field is tracked bya range finder 500 ft from the lift-off point. At the moment therange finder’s elevation angle is , the angle is increasingat the rate of 0.14 rad/min. How fast is the balloon rising atthat moment?

4

θRangeFinder 500 ft

y

0.14d

dt

rad/min

4 when

?dy

dt

4 when

Guided PracticeA hot-air balloon rising straight up from a level field is tracked bya range finder 500 ft from the lift-off point. At the moment therange finder’s elevation angle is , the angle is increasingat the rate of 0.14 rad/min. How fast is the balloon rising atthat moment?

4

θ500 ft

y

500 tany tan500

y

2

500 2 0.14 140dy

dt

2500 secdy d

dt dt

At the moment in question, the balloonis rising at the rate of 140 ft/min

Guided PracticeA police cruiser, approaching a right-angled intersection fromthe north, is chasing a speeding car that has turned the cornerand is now moving straight east. When the cruiser is 0.6 minorth of the intersection and the car is 0.8 mi to the east, thepolice determine with radar that the distance between them andthe car is increasing at 20mph. If the cruiser is moving at 60mphat the instant of measurement, what is the speed of the car?

x

y

60dy

dt ?

dx

dt

0

20ds

dt

Situation when x = 0.8, y = 0.6

x

ys

Guided Practice

x

y

60dy

dt ?

dx

dt

0

20ds

dt

Situation when x = 0.8, y = 0.6

2 2

12 2

2

ds dx dyx y

dt dt dtx y

2 2s x y x

ys

2 2

dx dyx ydt dtx y

Pythagorean Theorem:

Guided Practice

x

y

60dy

dt ?

dx

dt

0

20ds

dt

Situation when x = 0.8, y = 0.6

x

ys

2 2

0.8 0.6 6020

0.8 0.6

dx

dt

70dx

dt

At the moment in question,the car’s speed is 70 mph

Guided PracticeWater runs into a conical tank at the rate of 9 . The tankstands point down and has a height of 10 ft and a base radiusof 5 ft. How fast is the water level rising when the water is6 ft deep?

3ft min

10 ft

5 ft

x

y

3ft min9dV

dt

?dy

dt

when y = 6 ft

Guided PracticeWater runs into a conical tank at the rate of 9 . The tankstands point down and has a height of 10 ft and a base radiusof 5 ft. How fast is the water level rising when the water is6 ft deep?

3ft min

10 ft

5 ft

x

y

5

10

x

y

21

3V x y

Similar Triangles:

2

yx

21

3 2

yV y

3

12y

Guided PracticeWater runs into a conical tank at the rate of 9 . The tankstands point down and has a height of 10 ft and a base radiusof 5 ft. How fast is the water level rising when the water is6 ft deep?

3ft min

10 ft

5 ft

x

y

2312

dV dyy

dt dt

2

4

dyydt

At this moment, the water level is rising at about 0.318 ft/min.

29 64

dy

dt

10.318

dy

dt