Transcript Slide 1

Homework
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Homework Assignment #20
Review Section 3.11
Page 204, Exercises: 1 – 37 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
Consider 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?
dV
dh
dh 1 dV
V  lwh  Bh 
B


dt
dt
dt B dt
dh
1
7
  0.7   0.038 ft/min=
in/min
dt dV 0.7 18
15
dt
dV
7
 0.038 ft/min=
in/min
dt
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
Assume 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.
4 3
dv
dv
2
2 dr
V  r 
 4 r

 4  8  14 
3
dt
dt
dt r 8
3
dv
in
 3584
min
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
9. 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?
dy
dx
 0, x  3,  60  l 2  x 2  y 2  l  32  12  10
dt
dt
3

dl
dx
dy
dl 2 x dx
2l  2 x  2 y
 

 60   56.921mph
dt
dt
dt
dt 2l dt
10
y  1,
 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
13. 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?
27mi 1hr
d Isaac 
12 min  5.4mi
hr 60 min
32mi 1hr
d Sonya 
11min  5.86mi
hr 60 min
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
13. b) At what rate is the distance between them increasing at t = 12?
d x y 
2
2
 5.4 
2

 5.86

2
 7.974mi
dd
dx
dy
d  x  y  2d
 2x  2 y
dt
dt
dt
dx
dy
x y
5.4  27   5.86  32 
dd
dt
dt


 41.830mph
dt
d
7.974
2
2
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
17. 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?

 d
;
5 dt
 0.2; x  2; tan  
y
 y  2 tan 
x
dy
d
dy
2
 2sec 
 0.611 
 0.611 mi/min
dt
dt
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
Consider 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.
h  2   12  4 2  4, x  162  42  256  16  240  4 15
dh
dx
dh
dx
16  h  x  0  2h  2 x  h
 x
dt
dt
dt
dt
dx h dh
4
4



4

ft / s  1.033 ft / s
 
dt  x dt 4 15
15
2
2
2
dx
4
x  4 15 ft ; 
ft / s  1.033 ft / s
dt
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
25. 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?
1 2
dV 1 
dr
2 dh 
V  r h 
  2 rh   r

3
dt 3 
dt
dt 
dV 1
1
2
 2 10  20  2    10   2   1000 
dt 3
3
 1047.198cm3 / s


dV
 1047.198cm3 / s
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
29. 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.
x
dx
d
2
tan    x  y tan  
 y sec 
y
dt
dt
dx
 20000  sec 2 1.14   0.38   43581.688 ft / min
dt
43581.688 ft 1mi 60 min

 495.246mph
min
5280 ft hr
dx
 495.246mph
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
Assume 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.
PV b  C  V b
dP
bPV b 1

dt
Vb
dP
dV
dP
dV
 bPV b 1
 0 V b
 bPV b 1
dt
dt
dt
dt
1.2  8 

dV
bP dV


 20   1.92
dt
V dt
100
dP
 1.92 kPa/min
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 204
37. 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.
2
1 2
r 300
3
9 2
3 
2
V  r h  
 r  h  r   h  h
3
h 500
5
25
5 
1  9
3
dV
3
dh

V    h 2  h   h3 
3
 h2
3  25 
25
dt
25
dt
dh
25 dV
25
4



10


2.210

10


dt 9 h 2 dt 9  200 2
dh
 2.210 104 cm/min
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #21
Review Sections 3.1 – 3.11
Page 207, Exercises: 1 – 121 (EOO)
Chapter 3 Test next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company