Optimization
Min-Max Problems
Problem 1
A car rental agency rents 200 cars per day at a
rate of $30 per day. For each $1 increase in rate,
5 fewer cars are rented. At what rate should the
cars be rented to produce the maximum income?
What is the maximum income?
Answer to Problem 1
1. R(x) = (200 – 5x)(30 + x)
R(x) = 6000 + 50x – 5x2
R ’ (x) = 50 – 10x
R ‘ (x) = 0 50 – 10x = 0 x = 5
R “ (x) = –10 so f ” (x) is concave down
everywhere, so x = 5 is an amax
$35/day or 175 cars/day; $6125
Problem 2
A candy box is to be made out of a piece of cardboard
that measures 8 by 12 inches. Squares of equal size
will be cut out of each corner, and then the ends and
sides will be folded up to form a rectangular box.
What size square should be cut from each corner to
obtain a maximum volume?
Answer to Problem 2
V(x) = x(8 - 2x)(12 - 2x); 0 < x < 4
V(x) = 96x – 40x2 + 4x3
V ‘ (x) = 96 – 80x + 12x2
V ‘ (x) = 0 12x2 – 80x + 96 = 0
3x2 – 20x + 24 = 0
x = 5.097 in. or 1.569 in., but 0 < x < 4
therefore x = approx. 1.57 in
Problem 3
A family plans to fence in a rectangular patio area
behind their house. They have 120 feet of fence to
use. One side of the rectangle is the back of the
house. What should be the dimensions of the
rectangular region if they want to make the enclosed
patio area as large as possible.
Answer to Problem 3
A(x) = x(120 – 2x)
A(x) = 120x – 2x2
A ‘ (x) = 120 – 4x
A ‘ (x) = 0 120 – 4x = 0
; 30ft x 60ft = 1800 ft2
x = 30
Problem 4
A manufacturer of storage bins plans to produce some
open-top rectangular boxes with square bases. The
volume of each box is to be 125 cubic feet. material for
the base costs $6 per square foot, and material for the
sides costs $3 per square foot. Determine the dimensions
of the box that will minimize the cost of materials.
Answer to Problem 4
125
Cx   $6x  $3 4 x 2 
 x 
2
2 1500x1
C x 6x2 1500
6x
x
C ‘ (x) = 12x – 1500x–2




C ‘ (x) = 0
12x – 1500x–2 = 0
12x = 1500x–2
x3 = 125
12x3 = 1500
x=5
Problem 5
The owner of a warehouse decides to fence in an area
of 800 square feet behind the warehouse. He plans to
use the wall of the building as one of the four sides that
will enclose the rectangular area. He would like to use
the least amount of fencing necessary for the other
three sides. How many feet of fence will be needed.
Answer to Problem 5
800
f x  2x 
;20ft  40ft
x
80 ft. of fence

Problem 6
Given an isosceles triangle with equal sides of 3 inches
in length. Determine the measure of the angle between
the two equal sides which results in the largest area.
Answer to Problem 6
90 degrees
Problem 7
Construct a window in the shape of a semi-circle
over a rectangle (like many stained glass windows
in churches). If the distance around the outside of
the window is 12 feet, what dimensions will result
in the rectangle having the largest possible area?
Answer to Problem 7
P = 2r + πr + 2y
12 = 2r + πr + 2y
12 – 2r – πr = 2y
(12 – 2r – πr)/2 = y
A = 2ry
A = 2r (12 – 2r – πr)/2
A = r (12 – 2r – πr)
A = 12r – 2r2 – πr2
A = – 2r2 – πr2 + 12r
A’ = – 4r – 2πr + 12
0 = – 4r – 2πr + 12
4r + 2πr = 12
r = 12/(4 + 2π) = 1.66
Answers: The rectangle's dimensions would be 2.33
feet by 3 feet, so the area would be 6.99 square feet.
Problem 8
Ladder in the Hall Problem
A non-folding ladder is to be taken around a corner
where 2 hallways intersect at right angles. One hall
is 7' wide and the other hall is 5' wide. What is the
maximum length that the ladder can be so it will pass
around the corner?
Answer to Problem 8
Answer: 16.89 feet
Problem 9
During the Winter break, Jeremy Davis is enroute to his
uncle's house. Two policemen are stationed two miles
apart along the road, which has a posted speed limit of
55 mph. The first clocks him at 50 mph as he passes,
and the second policeman clocks him at 55 mph, but
pulls him over anyway. When he asks why he got pulled
over, the policeman responds, "It took you 90 seconds to
travel 2 miles." Why is he guilty of speeding according to
the Mean Value Theorem?
Answer to Problem 9
Answer: His average rate is 80 mph. There must have
been at least one point over the interval where the
instantaneous rate of change (velocity) must equal the
average rate of change (average speed).
Therefore, old lead foot Taylor must have traveled 80
mph at least once, and the policeman can ticket him.
Problem 10
The practice of shooting bullets into the air for whatever
purpose is extremely dangerous. Assuming that a hunting
rifle discharges a bullet with an initial velocity of 3000 ft/sec
from a height of 6 feet, answer the following questions:
(A) How high will the bullet travel at its peak?
(B) At what speed will the bullet be traveling when it
slams into the ground, assuming that it hits nothing in its path?
Answer to Problem 10
Answers:
(A) 140,631 feet (it takes 93.75 seconds to reach the
maximum height)
(B) 3000.064 ft/sec
Problem 11
To celebrate our 25th Anniversary in 2008, I
commissioned the construction of a four-inch tall box
made of precious metals to give to my wife. The jewelry
box will have rectangular sides and an open top. The
longer sides of the box were to be made of gold, at a cost of
$300 per in2; the shorter sides were to be made of platinum,
at a cost of $550 per in2. The bottom as to be made of
plywood, at a cost of 2 cents per in2.
What dimensions provide me with the lowest cost if I am
adamant that the box have a volume of 50 cubic inches?
Answer to Problem 11
Answer: C = 2400x + 4400y + xy(.02)
Dimensions are 4.787 in x 2.611 in x 4 in.
(The lowest cost for the box will be $23,000, do I still
need to buy a card?)
Problem 12
A builder is purchasing a rectangular plot of land with
frontage on a road for the purpose of constructing a
rectangular warehouse. Its floor area must be 300,000
square feet. Local building codes require that the building
be set back 40 feet from the road
and that there be empty buffer strips
of land 25 feet wide on the sides
and 20 feet in the back. Find the
overall dimensions of the parcel
of land and building which will
minimize the area of the land
parcel that the builder must buy.
Answer to Problem 12
Answer:
Dimensions of the building: 500' x 600'
Dimensions of the land: 550' x 660'