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

The objective function isArea = lw

l

w

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

… So, we rewrite the area formula as

… since we know p, we have

22

p lw

2( )2

p lA lw l

2880 2 4402lA l l l

A closer look at the AreaArea

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:

We know extrema exist where the derivative is zero …

440 2dA ldl

440 2 0440 2220

lll

SolutionSince this is the only critical point, it must be a maximum.

We can now solve for w, knowing that l = 220.

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.

2 880 2(220) 2202 2

p lw

General Guidelines1. 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?

SolutionArea: Objective function: minimize fencing.

Rewrite the area formula in terms of oneOf the variables.

Therefore:

216A l w

3 2Q w l

w

w

w

l l

Diagram

216wl

2 2216 648 2 648 23 2 l lQ l

l l l l

2

2

2( 324)' 0 18lQ ll

216 1218

w

3(12) 2(18) 72Q

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 single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

SolutionIntroduce notation: Length and width are ℓ and w. Length of wire used is p.1. Q = area = ℓw. – the objective function2. Since p = ℓ + 2w, we have

ℓ = p − 2w and so

However, remember that p = 800Q(w) = (p − 2w)(w) = 800w − 2w2

SolutionQ’ = 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 VolumeA 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 = x2h

To what constraint is the problem subjected?

S = x2 + 4xh = 108

Examples – Maximum VolumeA 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 AreaA 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 DistanceWhich 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