Homework

Homework Assignment #20 Review Section 3.11 Page 204, Exercises: 1 – 37 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Example, Page 204Consider a rectangular bathtub whose base is 18 ft2.1. How fast is the water level rising if water is filling the tub at a rate of 0.7 ft3/min?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

0.7

1

1 70.7 0.038 ft/min= in/min

18 15

70.038 ft/min= in/min

15

dV

dt

dV dh dh dVV lwh Bh B

dt dt dt B dtdh

dt

dV

dt

Example, Page 204Assume the radius r of a sphere is expanding at a rate of 14 in/min.5. Determine the rate at which the volume is changing with respect to time when r = 8 in.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

23 2

8

3

44 4 8 14

3

in3584 min

r

dv dr dvV r r

dt dt dt

dv

dt

Example, Page 2049. A road perpendicular to a highway leads to a farmhouse located 1 mile away. A car travels past the farmhouse at 60 mph. How fast is the distance between the farmhouse and car changing when the car is 3 miles past the intersection of the highway and road?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2 2 2 2 21, 0, 3, 60 3 1 10

322 2 2 60 56.921

2 10

dy dxy x l x y l

dt dt

dl dx dy dl x dxl x y mph

dt dt dt dt l dt

Example, Page 20413. Sonya and Isaac are in boats at the center of a lake. At t = 0, Isaac takes off, heading east at 27 mph. At t = 1 min, Sonya begins heading south at 32 mph.a) How far have Isaac and Sonya traveled at t = 12?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

27 112min 5.4

60min32 1

11min 5.8660min

Isaac

Sonya

mi hrd mi

hrmi hr

d mihr

Example, Page 20413. b) At what rate is the distance between them increasing at t = 12?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

222 2

2 2 2

5.4 5.86 7.974

2 2 2

5.4 27 5.86 3241.830

7.974

d x y mi

dd dx dyd x y d x y

dt dt dtdx dy

x ydd dt dt mphdt d

Example, Page 20417. A hot air balloon rising vertically is tracked by an observer located 2 mi from the lift-off point. At a certain moment, the angle between the observer’s line of sight and the horizontal is π/5, and it is changing at a rate of 0.2 rad/min. How fast is the balloon rising at that moment?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

; 0.2; 2; tan 2 tan5

2sec 0.611 0.611 mi/min

d yx y

dt x

dy d dy

dt dt dt

Example, Page 204Consider a 16-ft ladder sliding down a wall. The variable h is the height of the ladder’s top at time t and x is the distance from the wall to the ladder’s base.

21. Suppose h(0) = 12 and the top slides down the wall at a rate of 4 ft/s. Calculate x and dx/dt at t = 2s.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2 2

2 2 2

2 12 4 2 4, 16 4 256 16 240 4 15

16 0 2 2

4 44 / 1.033 /

4 15 15

44 15 ; / 1.033 /

15

h x

dh dx dh dxh x h x h x

dt dt dt dtdx h dh

ft s ft sdt x dt

dxx ft ft s ft s

dt

Example, Page 20425. Suppose that both the radius r and the height h of a circular cone are changing at the rate of 2 cm/s. How fast is the volume of the cone increasing when r = 10 and h = 20?

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2 2

2

3

3

1 12

3 3

1 12 10 20 2 10 2 1000

3 3

1047.198 /

1047.198 /

dV dr dhV r h rh r

dt dt dt

dV

dt

cm s

dVcm s

dt

Example, Page 20429. A plane traveling at 20,000 ft passes directly overhead at time t = 0. One minute later you observe the angle between the vertical and your line of sight to the plane is 1.14 rad and that the angle is changing at the rate of 0.38 rad/min. Calculate the velocity of the airplane.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

2

tan tan sec

20000 sec 1.14 0.38 43581.688 / min

43581.688 1 60min 495.246

min 5280

495.246

x dx dx y y

y dt dt

dxft

dtft mi

mphft hr

dxmph

dt

Example, Page 204Assume that the pressure P (in kilopascals) and volume V (in cm3) of an expanding gas are related by PVb = C, where b and C are constants.

33. Find dP/dt if b = 1.2, P = 8 kPa, V = 100cm3, and dV/dt = 20 cm3/min.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

1 1

1

0

1.2 820 1.92

100

1.92 kPa/min

b b b b b

b

b

dP dV dP dVPV C V bPV V bPV

dt dt dt dt

dP bPV dV bP dV

dt V dt V dt

dP

dt

Example, Page 20437. A water tank in the shape of a right circular cone of radius 300 cm and height 500 cm leaks water from the vertex at the rate of 10 cm3/min. Find the rate at which the water level is decreasing when it is 200 cm.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

22 2 2

2 3 2

422

4

1 300 3 3 9

3 500 5 5 25

1 9 3 33

3 25 25 25

25 2510 2.210 10

9 9 200

2.210 10 cm/min

rV r h r h r h h

h

dV dhV h h h h

dt dt

dh dV

dt h dt

dh

dt

Homework

Homework Assignment #21 Review Sections 3.1 – 3.11 Page 207, Exercises: 1 – 121 (EOO) Chapter 3 Test next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company