Calculus 1
WKST- Related Rates
Name:____________________________________
Date:_____________________________________
1.
Find the rate of change of the distance between the origin and a moving point on the graph of y  x 2  1
if dx dt  2 cm/sec.
x2  y2  r 2
x 4  3x 2  1  r
x
x 2  x 2  12  r 2
x 2  x 2  12
2.

 3x 2  1

12
r



1 2
1 4
dx dr
x  3x 2  1
 4x3  6x 

2
dt dt
r
Find the rate of change of the distance between the origin and a moving point on the graph of y  sin x
if dx dt  2 cm/sec.
x2  y2  r 2
x 2  sin x 2  r 2
x 2  sin x 2  r
3.
4
x
2
 sin 2 x

12

1 2
x  sin 2 x
2
r

1 2
2 x  2 sin x cos x
dx dr
 2 x  2 sin x cos x  

dt dt
x 2  sin 2 x

dr
dt
The radius (r) of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of
the area when.
dA
dr
A  r 2
 2r 
dt
dt
a.
r = 8 centimeters
b.
dA
 2 (8)( 4)
dt
dA
 64
dt
4.
dx
2
dt
dr
4x3  6x

dt
x 4  3x 2  1
r = 32 centimeters
dA
 2 (32)( 4)
dt
dA
 256
dt
The included angle of two sides of constant equal length s of an isosceles triangle is  . If  is
increasing at a rate of ½ radian per minute, find the rates of change of the area when:
1
A  ab sin C
2
a
1
( s )( s ) sin 
2
1
A  s 2 sin 
2
A
  6
dA 1 2  3  1 s 2 3
 
 s 
dt 2  2  2
8
dA 1 2
d
 s cos  
dt 2
dt
b.
  3
dA 1 2  1  1 s 2
 s   
dt 2  2  2 8
1
5.
The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the
volume when:
v
4 3
r
3
a.
dv
dr
 4r 2 
dt
dt
r = 9 inches
b.
dv
 4 (9) 2  (3)  972
dt
6.
dv
 4 (36) 2  (3)  15,552
dt
A spherical balloon is inflated with gas at a rate of 800 cubic centimeters per minute. How fast is the
radius of the balloon increasing at the instant the radius is:
v
4 3
r
3
a.
dv
dr
 4r 2 
dt
dt
30 cm
b.
800  4 (30) 2 
dr
dt
dr
dt
1
dr

18 dt
All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume
changing when each edge is:
dv
ds
 3s 2
dt
dt
v  s3
a.
2 cm
b.
dv
 3(2) 2 (6)
dt
dv
 72
dt
8.
60 cm
800  4 (60) 2 
2
dr

9 dt
7.
r = 36 inches
10 cm
dv
 3(10) 2 (6)
dt
dv
 1800
dt
All edges of a cube are expanding at a rate of 6 centimeters per second. Determine how fast the surface
area is changing when each edge is:
SA  6s 2
a.
dSA
ds
 12s 
dt
dt
2 cm
dSA
 12(2)(6)  144
dt
b.
10 cm
dSA
 12(10)(6)  720
dt
2
9.
1
The formula for the volume of a cone is v  r 2 h . Find the rates of change of the volume if dr/dt is 2
3
inches per minute and h = 3r when:
1
v  r 2  3r  r 3
3
a.
dv
dr
 3r 2 
dt
dt
r = 6 inches
b.
dv
 3 (6) 2  (2)  216
dt
10.
r = 24 inches
dv
 3 (24) 2  (2)  3456
dt
At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet
per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate
is the height of the pile changing when the pile is 15 feet high.
dv
3
 10; d  3h; r  h
dt
2
1
3 3
v  r 2 h 
h
3
4
dv 9 2 dh

h 
dt
4
dt
10 
9
dh
(15) 2 
4
dt
dh
8

dt 405
11.
A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the
tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the
water is 8 feet deep.
5 r

12 h
dv
 10
dt
dv 75 2 dh

h 
dt 432
dt
12.
r
10 
1
1  5h 
25 2
v  r 2 h      h 
h
3
3  12 
432
5h
12
75
dh
(8) 2 
432
dt
dh
9

dt 10
A trough is 12 feet long and 3 feet across the top. Its ends are isosceles triangles with altitudes of 3 feet.
v
a.
1
1
bhd  bh(12)  6bh
2
2
b=h
v  6h 2
dv
dh
 12h 
dt
dt
If water is being pumped into the trough at a rate of 2 cubic feet per minute, how fast is the water
level rising when the depth is 1 foot?
2  12(1) 
b.
b 3

h 3
dh
dt
dh 1

dt 6
If the water is rising at a rate of 3/8 inch per minute when h = 2, determine the rate at which the
water is being pumped into the trough.
dv
3
 12(2)   9
dt
8
3
13.
A ladder 25 feet long is leaning against a wall of a house. The base of the ladder is pulled away from
the wall at a rate of 2 feet per second.
a.
How fast is the top of the ladder moving down the wall when its base is 7 feet from the wall?
x 2  y 2  25 2
x 2  y 2  25 2
dy
dx
dy
 7 / 12
2x   2 y 
0
7 2  y 2  625
dt
dt
dt
y  24
dy
2(7)( 2)  2(24)
0
dt
b.
Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet
from the wall.
1
bh
2
1
A  xy
2
A
c.
 527
dA 1    7 
 (7)
  (24)( 2) 
dt 2   12 
 24
Find the rate at which the angle between the ladder and the wall of the house is changing when
the base of the ladder is 7 feet from the wall.
sin   7 / 25
  sin 1 (7 / 25)
14.
dA 1  dy
dx 
 x  y 
dt 2  dt
dt 

x
25
d
1
cos  

dt 25
sin  
cos sin 1 (7 / 25)
 ddt  251
d
1

rad / sec
dt 12
A baseball diamond has the shape of a square with sides 90 feet long. A player running from second
base to third base at a speed of 25 feet per second is 20 feet from third base. At what rate is the player’s
distance s from home plate changing?
x2  y2  r 2
20 2  90 2  r 2
r  10 85
x 2  90 2  r 2
dx
dr
2x 
 2r 
dt
dt
2(20)( 25)  2(10 85 )
dr
50

dt
85
dr
dt
*note- answer is negative because player is running to the left
15.
A baseball diamond has the shape of a square with sides 90 feet long. A player running from first
base to second base at a speed of 25 feet per second is 20 feet from second base. At what rate is the
player’s distance s from home plate changing?
x2  y2  r 2
70 2  90 2  r 2
r  10 130
x 2  90 2  r 2
dx
dr
2x 
 2r 
dt
dt
2(70)( 25)  2(10 130 )
dr
175

dt
130
dr
dt
*note- answer is negative because player is running to the left
4
16.
A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground.
When he is 10 feet from the base of the light,
a.
how fast is the tip of his shadow moving?
b.
how fast is the length of his shadow changing?
**Note- easier to do (b) first
15
6
x
y
x+y
b.
a.
x x  10

6
15
15 x  6 x  60
9 x  60
x
17.
20
3
10
25
5
3
3
x
6

x  y 15
15 x  6 x  6 y
9x  6 y
dx
dy
6
dt
dt
dx
9
 6(5)
dt
dx 10

dt
3
9
The combined electrical resistance R of R1 and R2 , connected in parallel, is given by:
1
1
1


R R1 R2
where R, R1 and R2 are measured in ohms. R1 and R2 are increasing at rates of 1 and 1.5 ohms per
second, respectively. At what rate is R changing when R1 = 50 ohms and R2 = 75 ohms?
R 1  R11  R21
1
1
1


R 50 75
150  3R  2 R
R  30
dR
 2 dR1
 2 dR2
  R1
  R2
dt
dt
dt
dR
 (30)  2
 (50)  2 (1)  (75)  2 (1.5)
dt
dR 5
 ohms / sec
dt 3
 R 2
5
18.
An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the
plane is 600 miles per hour. Find the rates at which the angle of elevation  is changing when the angle
is:
5
d
dx
tan  
sec 2  
 5 x  2
x
dt
dt
a.
 =30 degrees
b.
5
tan 30 
x
5
x
tan 30
 =60 degrees
2
sec 2 (30) 
d
 5 
 5
 600
dt
 tan 30 
d
 30 / sec
dt
19.
 = 75 degrees
tan 75 
x
5
x
5
tan 75
2
sec 2 (60) 
d
 5 
 5
 600
dt
 tan 60 
d
 90 / sec
dt
2
sec 2 (75) 
d
 5 
 5
 600
dt
 tan 75 
d
 111.962 / sec
dt
When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the
equation pV 1.3  k , where k is a constant. Find the relationship between the related rates dp dt &
dV dt .
dv
dP
P  1.3v
 v1.3
0
dt
dt
dv
dp 

v 3 1.3P  v   0
dt
dt 

3
20.
c.
5
tan 60 
x
5
x
tan 60
dv
 v dP

dt 1.3P dt
dP  1.3P dv

dt
v dt
A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost. The
ball’s shadow, caused by the light at the top of the lamppost, is moving along the level ground. How
fast is the shadow moving 1 second after the ball is released?
s (t )  16t 2  20
s (1)  16(1) 2  20  4  y
dy
v(t )  32t 
dt
v(1)  32
dx
dy
 240( y  20) 2 
dt
dt
20
y

x
x  12
xy  20 x  240
xy  20 x  240
x( y  20)  240
x  240( y  20) 1
dx
 240(4  20)  2   32  30 m / sec
dt
6
21.
A point moves along the curve y  x in such a way that the y-value is increasing at a rate of 2 units
per second. At what rate is x changing for each of the following value?
dy 1 1 2 dx
 x
dt 2
dt
a.
x=½
b.
11
2  
22
dx
1
dt
22.
1 2
dx
dt
x=1
2
c.
x=4
1 1 2 dx
1
2
dt
2
dx
4
dt
1 1 2 dx
4
2
dt
dx
 16
dt
The edges of a cube are expanding at a rate of 8 cm/sec. How fast is the surface area changing when
each edge is 6.5 cm?
SA  6s 2
dSA
ds
 12s
dt
dt
dSA
 12(6.5)(8)  624 cm 2 / sec
dt
23.
The cross section of a five-meter trough is an isosceles trapezoid with a two-meter lower base, a threemeter upper base, and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter
per minute. How fast is the water level rising when the water is 1 meter deep?
1
v  hb1  b2   d
2
dv
dh 15 dh
5  h
1
dt
dt 2 dt
v  dh2  b2 
2
dh 15 dh
3 b2
1 5 

1 
3h 
dt 2 dt
2 h
v  dh 2  
2 
2
dh 2

m / min
2
dt 25
15h
v  5h 
4
24.
A rotating beacon is located 1 km off a straight shoreline. If the beacon rotates at a rate of 3 revolutions
per minute, how fast (in kph) does the beam of light appear to be moving to a viewer who is ½ km down
the shoreline?
3
rev
rad 60 min
rad
 6

 360
min
min 1 hr
hr
x2  y2  r 2
2
1
12     r 2
2
5
r
2
1
tan   2
1
1
  tan (1 / 2)
x
tan  
1
tan   x
sec 2 
d dx

dt
dt
sec 2 (tan 1 (1 / 2))(360 ) 
450 
dx
dt
dx
dt
dx
 1413.717 kph
dt
7