Optimization and Related Rates Word Problems
1
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.
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
Optimization and Related Rates Word Problems
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 2 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.
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?
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 9 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
Optimization and Related Rates Word Problems
one-third the diameter of the base. If the sand is dumped at the rate of 2 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
2
increases at the rate of 3 cm /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?
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 pv
1.4
= 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 2 units per second at that
instant. Is the volume
decreasing or increasing?
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?
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?
54. An automobile traveling at a rate of 30 ft/sec is approaching an intersection.
Optimization and Related Rates Word Problems
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?
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 height H.
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?