1. A circular cylindrical container, open at the top and having a capacity of 24. cubic inches, is
to be manufactured. If the cost of the material used for the bottom of the container is three
times that used for the curved part and if there is no waste of material, find the dimensions
which will minimize the cost.
2. If a box with a square base and open top is to have a volume of 4 cubic feet, find the
dimensions that require the least material (neglect the thickness of the material and waster
in construction).
3. A page of a book is to have an area of 90 square inches, with 1-inch margins at the bottom
and sides and a .5-inch margin at the top. Find the dimensions of the page which will allow
the largest printed area.
4. Find the dimensions of the rectangle of maximum area that can be inscribed in a semicircle
of radius a, if two vertices lie on the diameter.
5. Find the maximum volume of a right circular cylinder that can be inscribed in a cone of
altitude 12 inches and base radius 4 inches if the axes of the cylinder and cone coincide.
6. Find the dimensions of the right circular cylinder of maximum volume that can be inscribed
in a sphere of radius a.
7. Prove that the rectangle of largest area having a given perimeter p is a square.
8. What number exceeds its square by the greatest amount?
9. A farmer wants to fence in 60,000 square feet of land in a rectangular plot along a straight
highway. The fence he plans to use along the highway costs $2 per foot, while the fence for
the other three sides costs $1 per foot. How much of each type of fence will he have to buy
in order to keep expenses to a minimum? What is the minimum expense?
10. A farmer wants to fence in 60,000 square feet of land in a rectangular plot and then divide it
in half with a fence parallel to one pair of sides. What are the dimensions of the rectangular
plot that will require the least amount of fence?
11. What are the dimensions of the base of the rectangular box of greatest volume that can be
constructed from 100 square inches of cardboard if the base is to be twice as long as it is
wide? (a) Assume the box has a top (b) Assume the box has no top.
12. A printed page has 1-inch margins at the top and bottom and .75 inch margins at the sides.
If the area of the printed portion is to be 48 square inches, what should the dimensions of
the page be to use the least paper?
13. A window is in the shape of a rectangle surmounted by a semicircle. Find the dimensions
when the perimeter is 12 meters and the area is as large as possible.
14. Find the dimensions of the rectangle of greatest area that can be inscribed in a circle of
radius r.
15. A rectangular plot of land containing 216 square meters is to be enclosed by a fence and
divided into two equal parts by another fence parallel to one of the sides. What dimensions
of the outer rectangle require the smallest total length for the two fences? How much fence
is needed?
16. An oil can is to be made in the form of a right circular cylinder to contain K cubic
centimeters. What dimensions of the can will require the least amount of material?
17. A container with a rectangular base, rectangular sides, and no top is to have a volume of
two cubic meters. The width of the base is to be 1 meter. When cut to size, material costs
$10 per square meter for the base and $5 per square meter for the sides. What is the cost
of the least expensive container?
18. An observatory with cylindrical walls and a hemispherical roof is to be constructed so that
its total volume of the cylindrical part will be 40,000 cubic feet. What dimensions will result
in the minimum total surface area for the wall and the roof?
19. A right circular cylinder in inscribed in a right circular cone so that the centerlines of the
cylinder and the cone coincide. The cone has height 6 and radius of base 3. Find the
volume and the dimensions of the cylinder that has maximum volume.
20. A manufacturer receives an order for oil cans that are to have a capacity of K cubic
centimeters. Each can is made from a rectangular sheet of metal by rolling the sheet into a
cylinder; the lids are stamped out from another rectangular sheet. What are the most
economical proportions of the can?
21. A man 6 ft. tall is walking at the rate of 3 ft/sec toward a street light 15 ft above the ground.
How fast is the length of his shadow changing? How fast is the tip of his shadow moving?
22. Show that if the surface area of a sphere changes at a constant rate, the volume changes
at a rate proportional to the radius.
23. At a certain instant an icicle in the shape of a right circular cone is 12 cm long and its
length is increasing at the rate of 0.5 cm/hr, while the radius of its base is 1 cm and is
decreasing at the rate of 0.05 cm/hr. Is the volume of the icicle increasing or decreasing at
that instant? At what rate?
24. Water is entering a conical reservoir 10 meters deep and 20 meters across the top at nine
cubic meters per minute. How fast is the water rising when it is 3 meters deep?
25. A conical water tank 10 ft deep and 6 ft across the top is leaking. If the water level is falling
at the rate of 2 ft/hr when the water is 3 ft deep, how fast is the water leaking at that instant?
26. Sand being dumped from a funnel forms a conical pile whose height is always one-third the
diameter of the base. If the sand is dumped at the rate of two cubic meters per minute, how
fast is the pile rising when it is 1 meter deep?
27. Pancake batter is poured into a pan to form a circular pancake whose area increases at
the rate of 3 cm2/sec. How fast is the radius increasing when the diameter of the
pancake is 10 cm?
28. The diagonal of a square increases at the rate of 3 m/sec. How fast is the area changing
when the side of the square is 6 meters?
29. The volume of a sphere is changing at a constant rate. Show that the surface area
changes at a rate which varies inversely as the radius.
30. Rain is falling into a cylindrical barrel at the rate of 20 cm3 per minute. If the radius of the
base is 18 cm, how fast is the water rising?
31. The water level in a conical reservoir 50 ft deep and 200 ft across the top is falling at the
rate of 0.002 ft /hr. How fast (in cubic feet per hour) is the reservoir losing water when the
water is 30 feet deep?
32. An icicle is in the shape of a right circular cone. At a certain point in time the height is 15
cm and is increasing at the rate of 1 cm/hr, while the radius of the base is 2 cm and is decreasing
at 1/10 cm/hr. Is the volume of ice increasing or decreasing at that instant? At what rate?
33. A boat floating several feet away from a dock is pulled in by a rope that is being wound up
by a windlass at the rate of 3 ft/sec. If the windlass is 4 ft above the level of the boat, how fast is
the boat moving through the water when it is 12 ft from the dock?
34. When a gas expands adiabatically (no energy change), its pressure and volume are related by
the equation pv1.4 = k, where k is a constant. Find the rate of change of volume at the instant
when p = 10 and v = 20, assuming that the pressure is decreasing two units per second at that
instant. Is the volume decreasing or increasing?
35. A spherical drop of water loses moisture by evaporation at a rate proportional to its surface
area. What can you say about its radius?
36. A pebble is dropped into a calm pool of water, causing ripples in the form of concentric
circles. If each ripple moves out from the center at a rate of 1 ft/sec, at what rate is the total
area of disturbed water increasing at the end of 4 sec?
37. A windlass is used to tow a boat to the dock. The rope is attached to the boat at a point 15
ft below the level of the windlass. If the windlass pulls in the rope at a rate of 30 ft per
minute, at what rate is the boat approaching the dock when there is 75 feet of rope out?
When there is 25 feet of rope out?
38. Water is flowing into a trough at a rate of 100 cubic centimeters per second. The trough
has a length of 3 m and cross sections in the form of a trapezoid, whose height of 50 cm,
whose lower base is 25 cm, and whose upper base is 1 m. At what rate is the water level
rising when the depth of the water is 25 cm?
39. Let A be the area of a circle of radius r. Find dA/dt. If dr/dt is constant, is dA/dt constant?
Explain why or why not.
40. A spherical balloon is inflated with gas at the rate of 20 cubic ft per minute. How fast is the
radius of the balloon increasing at the instant the radius is (a) 1 ft? (b) 2 ft?
41. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate
of 10 cubic ft/min. 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 it is 15 ft high?
42. A pebble is dropped into a calm pool of water, causing ripples in the form of concentric
circles. If each ripple moves out from the center at a rate of 1 ft/s, at what rate is the total
area of disturbed water increasing at the end of four s?
43. A windlass is used to tow a boat to the dock. The rope is attached to the boat at a point 15
ft below the level of the windlass. If the windlass pulls in the rope at a rate of 30 ft/min, at
what rate is the boat approaching the dock when there is 75 ft of rope out? When there is
25 ft of rope out?
44. A tank is in the form of an inverted cone having an altitude of 15 m and a radius of 4 m.
Water is flowing into the tank at the rate of 2 m3/min. How fast is the water level rising
when the water is 5 m deep?
45. Two cars, one going due east at the rate of 90 km/hr and the other going due south at the
rate of 60 km/hr, are traveling toward the intersection of two roads. At what rate are the two
cars approaching each other at the instant when the first car is .2 km and the second car is
.15 km from the intersection?
46. A spherical balloon is being inflated so that its volume is increasing at the rate of 5m3/min.
At what rate is the diameter increasing when the diameter is 12 m?
47. A spherical snowball is being made so that its volume is increasing at the rate of 8 ft3/min.
Find the rate at which the radius is increasing when the snowball is 4 ft in diameter.
48. Suppose that when the diameter is 6 ft, the snowball in exercise 72 stopped growing and
started to melt at the rate of (1/4) ft3/min. Find the rate at which the radius is changing when
the radius is 2 ft.
49. Sand is being dropped at the rate of 10 m3/min onto a conical pile. If the height of the pile
is always twice the base radius, at what rate is the height increasing when the pile is 8 m
high?
50. Suppose that a tumor in a person's body is spherical in shape. If, when the radius of the
tumor is 0.5 cm, the radius is increasing at the rate of 0.001 cm per day, what is the rate of
increase of the volume of the tumor at that time?
51. A trough is 12 ft long and its ends are in the form of inverted isosceles triangles having an
altitude of 3 ft and a base of 3 ft. Water is flowing into the trough at the rate of 2 ft3/min. How
fast is the water level rising when the water is 1 ft deep?
52. A stone is dropped into a still pond. Concentric circular ripples spread out, and the radius
of the disturbed region increases at the rate of 16 cm/sec. At what rate does the area of the
disturbed region increase when its radius is 4 cm?
53. Oil is running into an inverted conical tank at the rate of 3p m3/min. If the tank has a radius
of 2.5 m at the top and a depth of 10 m, how fast is the depth of the oil changing when it is
8m?
54. An automobile traveling at a rate of 30 ft/sec is approaching an intersection. When the
automobile is 120 ft from the intersection, a truck traveling at the rate of 40 ft/sec crosses the
intersection. The automobile and the truck are on roads that are at right angles to each
other. How fast are the automobile and the truck separating 2 sec after the truck leaves the
intersection?
55. Show that if the volume of a balloon is decreasing at a rate proportional to its surface area,
the radius of the balloon is shrinking at a constant rate.
56. In a lake a predator fish feeds on a smaller fish, and the predator population at any time is
a function of the number of small fish in the lake at that time. Suppose that when there are x
small fish in the lake, the predator population is y, and y = (1/60,000) x2 - (1/100)x + 40. If
the fishing season ended t weeks ago, x = 300 t + 375. At what rate is the population of the
predator fish growing 10 weeks after the close of the fishing season? Do not express y in
terms of t, but use the chain rule.
57. A funnel in the form of a cone is 10 in. across the top and 8 in. deep. Water is flowing into
the funnel at the rate of 12 in3/sec and out at the rate of 4 in3/sec. How fast is the surface of
the water rising when it is 5 in. deep?
58. Find the dimensions of the right circular cylinder of largest volume that can be inscribed in
a right circular cone of radius R and altitude H.
59. A truck is 250 miles due east of a sports car and is traveling west at a constant speed of 60
miles per hour. Meanwhile, the sports car is going north at 80 miles per hour. When will the
truck and the car be closest to each other? What is the minimum distance between
them?
60. A stained glass window in the form of an equilateral triangle is built on top of a rectangular
window. The rectangular part of the window is of clear glass and transmits twice as much
light per square foot as the triangular part, which is made of stained glass. If the entire
window has a perimeter of 20 feet, find the dimensions (to the nearest foot) of the window
that will admit the most light.
61. An environmental study of a certain community indicates that there will be
units of a harmful pollutant in the air when the population is thousand. The population is
currently 30,000 and is increasing at a rate of 2,000 per year. At what rate is the level of air
pollution increasing currently?
62. Hospital officials estimate that approximately 2()5900Nppp=++people will seek
treatment in the emergency room each year if the population of the community is
thousand. The population is currently 20,000 and is growing at the rate of 1,200 per year.
At what rate is the number of people seeking emergency room treatment increasing?
p
63. A ladder 13 feet long rests against a vertical wall and is sliding down the wall at the rate of
3 ft/s at the instant the foot of the ladder is 5 feet from the base of the wall. At this instant,
how fast is the foot of the ladder moving away from the wall?
64. A car traveling north at 40 miles per hour and a truck traveling east at 30 miles per hour
leave an intersection at the same time. At what rate will be distance between them be
changing 3 hours later?
65. A person is standing at the end of a pier 12 feet above the water and is pulling in a rope
attached to a rowboat at the waterline at the rate of 6 feet of rope per minute. How fast is
the boat moving in the water when it is 16 feet from the pier?
66. One end of a rope is fastened to a boat and the other end is wound around a windlass
located on a dock at a point 4 m above the level of the boat. If the boat is drifting away from
the dock at the rate of 2 m/ min, how fast is the rope unwinding at the instant when the
length of the rope is 5 meters?
67. A certain medical procedure requires that a balloon be inserted into the stomach and then
inflated. Model the shape of the balloon by a sphere of radius r. If f is increasing at the rate
of 0.3 cm/min, how fast is the volume changing when the radius is 4 cm?
68. Model a water tank by a cone 40 ft high with a circular base of radius 20 feet at the top.
Water is flowing into the tank at a constant rate of 80 ft3/min. How fast is the water level
rising when the water is 12 feet deep? Give your answer to the nearest hundredth of a foot
per minute.
69. A block of ice in the shape of a cube originally having volume 1000 cm3 is melting in such a
way that the length of each of its edges is decreasing at the rate of 1 cm/hr. At what rate is
its surface area decreasing at the time its volume is 27 cm3? Assume that the block of ice
maintains its cubical shape.