Section 6.6Related Rates

1. When a circular plate of metal is heated in an oven its radius increases at a rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?

2A rdA dr

2 rdt dt

d0.01

A2 50

dt

dA

dt

dr0.01

dt r 50

2. The length L of a rectangle is decreasing at the rate of 2 cm/sec and the width W is increasing at the rate of 2 cm/sec. When L = 12 cm and W = 5 cm, find the rates of change of:(a) the area

A LW

dA dL dWW L

dt dt dt

2A

dt5 1

d2 2

dA14

dt

dL2

dt dW

2dt

L 12 W 5

2. The length L of a rectangle is decreasing at the rate of 2 cm/sec and the width W is increasing at the rate of 2 cm/sec. When L = 12 cm and W = 5 cm, find the rates of change of:(b) the perimeter

P 2L 2W

dP dL dW2 2

dt dt dt

dP2

t22

d2

The perimeter does not change

dL2

dt dW

2dt

L 12 W 5

2. The length L of a rectangle is decreasing at the rate of 2 cm/sec and the width w is increasing at the rate of 2 cm/sec. When L = 12 cm and w = 5 cm, find the rates of change of:(c) the lengths of the diagonals of the rectangle?

2 2 2D L W

dD dL dW2D 2L 2W

dt dt dt

dD dL dWD L W

dt dt dt

2 2 2D 12 5

D 13

11 2d

5t

23D

d2

dD 14

dt 13

dL2

dt dW

2dt

L 12 W 5

3. Sand falls from a conveyor belt onto a conical pile at the rate of 10 cubic feet per minute. The radius of the base of the pile is always equal to ½ of the altitude. How fast is the altitude of the pile increasing when the pile is 5 feet high?

1r h

dh

dhFind

dtt51

2

V0

d21

V r h3

231 1

V h V h3 1

1h

2 2

2dV 1 dhh

dt 4 dt

2510

4 t

1 dh

d

dh 8 ft0.509

dt 5 min

4. A point moves on the curve so that its y-coordinate increases at a constant rate of 6 meters per second. a. At what rate is the x-coordinate changing when x = 4

meters? b. What is the slope of the curve when x = 4 m?

2 23x y 12

xd

yt

6y

6d

4 2 23x y 12

dx dy6x 2y 0

dt dt

dx

dt4 2 6 066

dx m3

dt sec

dy6dtm 2

dx 3dt

5. A spherical balloon is inflated with gas at the rate of 100 cubic feet per minute. a. Assuming that the gas pressure remains constant, how fast

is the radius of the balloon increasing at the instant when the radius is three feet?

b. How fast is the surface area increasing?

dV100 r 3

dt

34V r

3

2dV dr4 r

dt dt

2100 3

dr4

dt

25 ft0.884

9

d

t min

r

d

2S 4 r

dS dr8 r

dt dt

25

9

dS8

dt3

2dS 200 ft66.667

dt 3 min

6. A point P moves from left to right along the curve at a constant horizontal speed of 3 units per second. How fast does the y-coordinate of P increase at the moment when P passes through (2, 4)?

2y x

2y x xd

yt

4x

3d

2

dy dx2x

dt dt

dy

dt32 2

dy12

dt

7. The length of the base of a right triangle is increasing at the rate of 2 inches per minute. At the same time, the height of the triangle is decreasing in such a way that the length of the hypotenuse remains 10 inches. When the length of the base is 6 inches, how quickly is the height of the triangle changing?

2 2 2x y z zdx

2 x 6dt

10 dz

y 8 0dt

dx dy dzx y z

dt dt dt

dy6

d8 02

t01

dy1.5

dt