Optimization Problems
Section 4-4
Example
 What is the maximum area of a
rectangle with a fixed perimeter of
880 cm?
In this instance we want to optimize (maximize)
the area of a rectangle.
Solution
 Draw a rectangle
w
The objective function is
Area = lw
l
Solution
 This is a function of two variables, so we need to use the
constraint that the perimeter is fixed. Since p = 2ℓ + 2w,
we have
p  2l
w
2
… So, we rewrite the area formula as
p  2l
A  lw  l (
)
2
… since we know p, we have
 880  2l 
2
Al

440
l

l

2 

A closer look at the Area
Area
Graph of the
possible area of
the rectangle
dependent on its
length, l.
Solution
 Now we have A as a function of “l” alone (p is constant).
The natural domain of this function is [0, p/2]. Both of
these endpoints would result in a degenerate rectangle.
Let’s take derivatives:
dA
 440  2l
dl
We know extrema exist where the derivative is zero …
440  2l  0
440  2l
220  l
Solution
Since this is the only critical point, it must be a maximum.
We can now solve for w, knowing that l = 220.
p  2l 880  2(220)
w

 220
2
2
Conclusion: Maximum area is given by A = lw = 220∙220 =48,400
Also, note that maximum area of a rectangle is given by a square.
General Guidelines
1. Understand the Problem. What is known? What is unknown? What are
the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols.
5. If Q is a function of more than one “decision variable”, use the given
information to eliminate all but one of them.
6. Find the absolute maximum (or minimum, depending on the problem) of
the function on its domain. Do this by taking the derivative of the
objective function. Watch for EXTRANEOUS solutions (0 or negative
values).
Another Example
 A 216m2 rectangular pea patch is to be enclosed by a
fence and divided into two equal parts by another fence
parallel to one of its sides. What dimensions for the
outer rectangle will require the smallest total length of
fence? How much fence will be needed?
Solution
Area: A  l  w  216
Objective function: minimize fencing.
Diagram
w
Q  3w  2l
Rewrite the area formula in terms of one
Of the variables. w  216
l
l
216
648 2l 2 648  2l 2
Q  3
 2l 


l
l
l
l
2(l 2  324)
Q' 
 0  l  18
2
l
Therefore:
w
w
w 
216
 12
18
Q  3(12)  2(18)  72
l
Yet another example
 A rectangular plot of farmland will be bounded on one
side by a river and on the other three sides by a singlestrand electric fence. With 800 m of wire at your
disposal, what is the largest area you can enclose, and
what are its dimensions?
Solution
Introduce notation: Length and width are ℓ and
w. Length of wire used is p.
1. Q = area = ℓw. – the objective function
2. Since p = ℓ + 2w, we have
ℓ = p − 2w and so
However, remember that p = 800
Q(w) = (p − 2w)(w) = 800w − 2w2
Solution
Q’ = 800 − 4w = 0
 derivative is zero when w = 800/4 = 200
 Substitute back into the objective function
Q(w) = 800w – 2w2
Q (200)= 800(200) − 2 (200)2
= 80,000 m2
Therefore, the maximum area that can be enclosed by 800
meters of fencing, given the original constraints, is
80,000 m2.
Examples – Maximum Volume
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 of maximum volume?
Sketch a diagram
What do you wish to optimize (maximize)?
V = x 2h
To what constraint is the problem subjected?
S = x2 + 4xh = 108
Examples – Maximum Volume
A sheet of cardboard 3 ft. by 4 ft. will be made
into a box by cutting equal-sized squares from
each corner and folding up the four edges.
What will be the dimensions of the box with
largest volume ?
Sketch a diagram
What do you wish to optimize (maximize)?
V = l*w*h
To what constraint is the problem subjected?
V= (3-2x)(4-2x)x
Examples – Minimum Area
A rectangular poster is to contain 24 square inches of print.
Margins on the top and the bottom of the page are 1½ inches,
and the margins on the left and right are to be 1 inch. What
should the dimensions of the page be so that the least amount
of paper is used?
Sketch a diagram
What do you wish to optimize (minimize)?
A = (y+3)(x+2)
To what constraint is the problem subjected?
24 = xy
Examples – Minimum Distance
Which points on the graph of y = 4 – x2 are closest to the point
(0,2)
Sketch a diagram
What do you wish to optimize (minimize)?
d=√(x-0)2+(y-2)2
To what constraint is the problem subjected?
y = 4 - x2