Compute the rate of change of one quanti ty in terms of ...

[Pages:7]Math 103 ? Rimmer 3.10 Related Rates

Goal :

Compute the rate of change of one quantity in terms of the rate of

change of another quantity (which may be more easily measured).

ex.1 A cylindrical tank with radius 5 m is being filled with water

Math 103 ? Rimmer 3.10 Related Rates

at a rate of 3 m3 / min . How fast is the height of the water increasing?

dV = 3 dt

Find dh . dt

5 m.

Vcylinder = r 2h

V = 25 h since r = 5 (constant)

5 m.

h

dV = 25 dh

dt

dt

dh = 3 m./min.

dt 25

ex.2

Math 103 ? Rimmer 3.10 Related Rates

A plane flying horizontally at an altitude of 1 mi. and a speed of 500 mi./hr.

passes directly over a radar station. Find the rate at which the distance from the

plane to the station is increasing when it is 2 mi. away from the station.

dx = 500

dt

x

Find dz . dt

1 mi. z

12 + x2 = z2 2x dx = 2z dz dt dt

"instant snapshot"

x= 3

2 3 500 = 2 2 dz dt

1

z=2

dz = 1000 3 = 250 3 mi./hr.

dt

4

ex.3

Math 103 ? Rimmer 3.10 Related Rates

At noon, ship A is 150 km. west of ship B. Ship A is sailing east at 35 km./hr.

and ship B is sailing north at 25 km./hr. How fast is the distance between the

ships changing at 4:00 pm?

"instant snapshot"

Find dz . dt

z

y

dy = 25 dt

x = 150 - 35(4) x = 10

x = 150 -140

A

x

x is decreasing

B

dx

dx < 0

= -35 dt

dt

x2 + y2 = z2

10 101 100 10

y = 25(4) y = 100

z = 1002 +102 z = 10, 000 +100 z = 10,100

2x dx + 2 y dy = 2z dz dt dt dt

(10)(-35) + (100)(25) = 10,100 dz dz = -350 + 2500 = 2150 =

dt dt

10,100

10,100

215 km./hr. 101

ex.4

Math 103 ? Rimmer 3.10 Related Rates

A street light is mounted at the top of a 15 ft. tall pole. A man 6 ft. tall walks

away from the pole with a speed of 5 ft./s. along a straight path. How fast is

the tip of his shadow moving when he is 40 ft. from the pole?

s = the distance to the tip of the shadow s = x + y ds = dx + dy

dt dt dt

known unknown

Find ds . dt

dx = 5 dt

15 6

x

y

Use similar triangles :

15 = 6 15y = 6x + 6 y

x+ y y

9y = 6x

y=2x 3

dy = 2 dx = 2 (5) = 10

dt 3 dt 3

3

ds = 5 + 10 = 15 +10 = 25 ft./s.

dt

33 3

ex.5

Math 103 ? Rimmer 3.10 Related Rates

Two sides of a triangle have lengths 12 m. and 15 m. The angle between them

is increasing at a rate of 2 / min . How fast is the length of the third side

increasing when the angle between sides of fixed length is 60 ? Find dx .

dt

2 = radians 90

Use Law of Cosines :

15

x

c2 = a2 + b2 - 2ab cos ( is the opposite c)

x2 = 152 +122 - 2(15)(12) cos ( )

12

"instant snapshot"

2x

x2 dx

= =

369 - 360

360sin (

cos( ) ) d

dx

=

360 sin

(

)

d

dt

dt

dt dt

2x

15

3

12

x

x=

369

-

360

cos

(

3

)

= 189

= 3 21

dx = 360

3

2 90

2

360 =

3

2 90

dt 2 3 21

2 3 3 7

dx = m./min.

dt 3 7

ex.6 Water is leaking out of an inverted conical tank at a rate of 10,000 cm3 /min. at the same time that water is being poured into the tank at a constant rate. The tank has height 6 m. and the diameter at the top is 4 m. If the water level is

Math 103 ? Rimmer 3.10 Related Rates

Let W = rate water is being pumped in

rising at a rate of 20 cm./min. when the height of the water is 2 m., find the

Find W .

rate at which the water is being pumped into the tank.

2

Vcone

=

3

r2h

V

=

3

h 3

2

h

V = h3

27

dV = h2 dh

r

dt 9 dt

6

h

"instant snapshot"

h 2 200 3

W -10, 000 = (200)2 (20)

9

r =2 h6

6r = 2h 6

r=h 3

2 r

h

h = 200

W = 10, 000 + 800, 000 cm.3 / min .

9

dV

= W -10, 000

dt rate

rate

in

out

dh = 20 dt

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