11DIFFERENTIATION - Department of Mathematics

49. O IL S PILLS In calm waters, oil spilling from the ruptured
hull of a grounded tanker spreads in all directions. If
the area polluted is a circle and its radius is increasing
at a rate of 2 ft/sec, determine how fast the area is
increasing when the radius of the circle is 40 ft.
50. Two ships leave the same port at noon. Ship Asails
north at 15 mph, and ship B sails east at 12 mph. How
fast is the distance between them changing at 1 P.M.?
51. Acar leaves an intersection traveling east. Its position
t sec later is given by x t 2 t ft. At the same time,
another car leaves the same intersection heading north,
traveling y t 2 3t ft in t sec. Find the rate at which
the distance between the two cars will be changing
5 sec later.
52. At a distance of 50 ft from the pad, a man observes a
helicopter taking off from a heliport. If the helicopter
lifts off vertically and is rising at a speed of 44 ft/sec
when it is at an altitude of 120 ft, how fast is the distance
between the helicopter and the man changing at that instant?
53. Aspectator watches a rowing race from the edge of a
river bank. The lead boat is moving in a straight line
that is 120 ft from the river bank. If the boat is moving
at a constant speed of 20 ft/sec, how fast is the boat
moving away from the spectator when it is 50 ft past her?
54. Aboat is pulled toward a dock by means of a rope which
is wound on a drum that is located 4 ft above the bow
of the boat. If the rope is being pulled in at the rate of
3 ft/sec, how fast is the boat approaching the dock when
it is 25 ft from the dock?
55. Assume that a snowball is in the shape of a sphere. If
the snowball melts at a rate that is proportional to its
surface area, show that its radius decreases at a constant
rate.
Hint: Its volume is V (4/3)r 3 , and its surface area is S 4r 2 .
56. B LOWING S OAP B UBBLES Carlos is blowing air into a soap
bubble at the rate of 8 cm 3 /sec. Assuming that the bubble
is spherical, how fast is its radius changing at the instant
of time when the radius is 10 cm? How fast is the surface
area of the bubble changing at that instant of time?
11.6 IMPLICIT DIFFERENTIATION AND RELATED RATES 769
57. C OAST G UARD P ATROL S EARCH M ISSION The pilot of a Coast
Guard patrol aircraft on a search mission had just spotted
a disabled fishing trawler and decided to go in for a
closer look. Flying at a constant altitude of 1000 ft and
at a steady speed of 264 ft/sec, the aircraft passed directly
over the trawler. How fast was the aircraft receding from
the trawler when it was 1500 ft from it?
1000 ft
58. Acoffee pot in the form of a circular cylinder of radius
4 in. is being filled with water flowing at a constant rate.
If the water level is rising at the rate of 0.4 in./sec, what
is the rate at which water is flowing into the coffee pot?
h
59. A6-ft tall man is walking away from a street light 18 ft
high at a speed of 6 ft/sec. How fast is the tip of his
shadow moving along the ground?
60. A20-ft ladder leaning against a wall begins to slide. How
fast is the top of the ladder sliding down the wall at the
instant of time when the bottom of the ladder is 12 ft
from the wall and sliding away from the wall at the rate
of 5 ft/sec?
Hint: Refer to the adjacent figure. By the Pythagorean theorem,
x 2 y 2 400. Find dy/dt when x 12 and dx/dt 5.
5 ft/sec
20-ft ladder
x
Wall
y