Optimization Problems
Applied Minimum &
Maximum Problems
Section 3.7
“Geometric Problems”
Area / Perimeter / Volume /
Surface Area
Example
• A metal box (without a top) is to be
constructed from a square sheet of metal
10 in. on a side by first cutting square
pieces of the same size from the corners
of the sheet and then folding the sides up.
Find the dimensions of the box with the
largest volume that can be constructed.
Example
• A manufacturer wants to design
an open box having a square
base and a surface area of 108
square inches. What
dimensions will produce a box
with maximum volume?
Example
• What is the radius of a cylindrical soda
can with volume of 512 cubic inches that
will use the minimum material?
“Shortest Route”
Usually has 2 different speeds!
Example
• A swimmer is at a point 500 m. from the
closest point on a straight shoreline. She
needs to reach a cottage located 1800 m.
down shore from the closest point. If she
swims 4 m/sec and walks 6 m/s, how far
from the cottage should she come ashore
so as to arrive at the cottage in the
shortest time?
Example
• A girl standing on a straight road wants to run
down the road and then diagonally through a field
to her car, which she left 1 mile from the nearest
point on the road. That point on the road is 2
miles from where she is now standing. If she can
run 5 mph on the road, but only 3 mph through the
field, at what point should she leave the road and
cut through the field in order to get to her car as
quickly as possible?
Example
• It is commonly known that homing pigeons fly
faster over land than over water. Assume that
instead of flying 10 meters per second (as over
land), they fly only 9 meters per second over
water. If a pigeon is located at the edge of a river
500 meters wide and must fly to its nest 1300
meters away on the opposite edge of the river,
what path would minimize its flying time?
“Cost or Profit”
Example
• It costs a bus company $125 to run a bus on a
certain tour, plus $15 per passenger. The capacity
of the bus is 20 persons and the company charges
$35 per ticket if the bus is full. For each empty
seat; however, the company increases the ticket
price by $2 per person. For maximum profit, how
many empty seats would the company prefer?
(Must use calculus).
Example
• A bookstore can buy books from a publisher at a
cost of $6/book and has been selling 200 of the
books per month to the public at $30/copy. The
bookstore is planning to lower its price per book
to stimulate sales, and estimates that for each $2
reduction in price, 20 more books will be sold per
month. At what price should the bookstore sell the
books in order to generate the greatest profit?
Miscellaneous
Example
• What is the largest possible product of two
nonnegative numbers whose sum is 1?
Example
• At noon a sailboat is 20 km south of a
freighter. The sailboat is traveling east at
20 km/hr, and the freighter is traveling
south at 40 km/hr. If visibility is 10 km,
could the people on the 2 ships ever see
each other?
Example
• A plane flying due north at 750 mph flies
directly over a car traveling due west at 80
mph. If the distance between the car and
the plane is changing at a rate of 200 mph 1
minute later, what is the altitude of the
plane?
Project Problem 1
• If you plan to make an open-topped box out
of a sheet of tin 24” wide by 45” long by
cutting congruent squares out of each corner
and then bending up and soldering the
resulting flaps at the corners, what should
be the dimensions of the box in order to
have the largest volume?
Project Problem 2
• A landowner wishes to use 2000 meters of
fencing to enclose a rectangular region.
Suppose one side of the property lies
along a stream and thus doesn’t need to
be fenced in. What should the lengths of
the sides be in order to maximize the
area?
Project Problem 3
• A forest ranger is in a forest 2 miles from
a straight road. A car is located 5 miles
down the road. If the ranger can walk 3
mi/hr in the forest and 4 mi/hr along the
road, toward what point on the road
should the ranger walk in order to
minimize the time needed to walk to the
car?
Project Problem 4
• An island is at a point A, 6 km offshore from the
nearest point B on a straight beach. A woman on
the island wishes to go to point C, 9 km down the
beach from B. The woman can rent a boat for
$15/km and travel by water to a point P between B
and C, and then she can hire a car with a driver
with a cost of $12/km and travel a straight road
from P to C. Find the least expensive route for
her.
Project Problem 5
• A cable TV co. wishes to place an amplifier
station at a point on a street & run wires from
the station to 3 houses. One house is adjacent
to the street, & 2 are 50 ft from the street.
Where should the station be located in order to
minimize the total length of wire required to
service all three houses? ** The distance along
the street between the houses is 200 feet.
Project Problem 6
• The Spice-of-Life Company is preparing
to create shipping crates. The co. wishes
the volume of each crate to be 6 cu. Ft, w/
the crate’s base to be square b/w 1 and 2
ft on a side. Assume the material for the
bottom costs $5, the sides $2, and the top
$1 / sq. ft. Find the dimensions that yield
the minimum cost.
Project Problem 7
• A plane flying 400 mph crosses at right
angles 10 miles over a car going 50 mph.
Two hours later how fast is the distance
between the plane and the car increasing?