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

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

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