Optimization

Example 12. Find two real numbers with a difference of 8 such that the sum of
the squares of the two numbers is a minimum.

Let’s begin by letting x and y re present the numbers we seek. Next, let’s play
a bit as we did in the previous example. Try x = 9 and y = 1. The difference of
these two numbers is certainly 8. The sum of the squares of these two numbers is
S = 9² + 1² = 82. Let’s put this result in tabular form.

For a second guess, select x = 8 and y = 0. The difference is x−y = 8−0 = 8, but
this time the sum of the squares is S = 8² + 0² = 64. For a third guess, try x = 7 and
y = −1. Again, the difference is x − y = 7 − (−1) = 8, but the sum of the squares is
now S = 7² + (−1)² = 50. Let’s add these results to our table.

Thus far, the pair that minimizes the sum of the squares is x = 7 and y = −1.
However, could there be another pair with a difference of 8 and the sum of the squares
is smaller than 50? Experiment further to see if you can best the current minimum of
50.

Let’s try an analytical approach. Our first constraint is the fact that the difference
of the two numbers must equal 8. This is easily expressed as

x − y = 8. (13)

Next, we’re asked to minimize the sum of the squares of the two numbers. This requires
that we find a formula for the sum of the squares. Let S represent the sum of the squares
of x and y. Thus,

S = x² + y². (14)

Note that S is a function of two variables. We can eliminate one of the variables by
solving equation (13) for x,

x = y + 8, (15)

then substituting this result in equation (14).

S = (y + 8)² + y².

Expand and simplify.

S = 2y² + 16y + 64 (16)

Compare S = 2y²+16y+64 with the general quadratic S = ay²+by+c and note that
a = 2 and b = 16. Thus, the plot of S versus y will be a parabola that opens upward
(see Figure 4) and the minimum value of S will occur at the vertex. The y-coordinate
of the vertex is found with

Figure 4. Plotting the sum of the
squares S versus y. The minimum S,
32, occurs at the vertex, (−4, 32).

Thus, the first number we seek is y = −4. We can find the second number by substituting
y = −4 in equation (15).

x = y + 8 = (−4) + 8 = 4.

Hence, the numbers we seek are x = 4 and y = −4. Note that the difference of these two
numbers is x−y = 4−(−4) = 8 and the sum of their squares is S = (4)²+(−4)² = 32,
which is smaller than the best result found in our tabular experiment above. Indeed,
our work show that this is the smallest possible value of S.

Alternatively, you can find S by substituting y = −4 in equation (16). We’ll leave
it to our readers to verify that this also gives a minimum value of S = 32.

Let’s look at another application.

Example 17. Mary wants to fence a rectangular garden to keep the deer from
eating her fruit and vegetables. One side of her garden abuts her shed wall so she will
not need to fence that side. However, she also wants to use material to separate the
rectangular garden in two sections (see Figure 5). She can afford to buy 80 total feet
of fencing to use for the perimeter and the section dividing the rectangular garden.
What dimensions will maximize the total area of the rectangular garden?

Figure 5. Mary’s rectangular garden
needs fencing on three sides and also for
the fence to divide the garden.

Again, before we take an algebraic approach , let’s just experiment. Note that we’ve
labeled the width with the letter x and the height with the letter y in our sketch of the
garden in Figure 5.

There is a total of 80 feet of fence material. Suppose that we let y = 5 ft. Because
there are three sides of length y = 5 ft, we’ve used 15 feet of material. That leaves 65
feet of material which will be used to fence the width of the garden. That is, the width
is x = 65 ft. Thus, the dimensions of the garden are x = 65 ft by y = 5 ft. The area
equals the product of these two measures, so A = 325 ft². Let’s put this result into a
table.

Suppose instead that we let the height be y = 10 ft. Again, there are three sections
with this length, so this will take 30 ft of material. That leaves 50 ft of material, so the

width x = 50 ft. The area is the product of these two measures, so A = 500 ft². As
a third experiment, let the height y = 15 ft. Subtracting three of these lengths from
80 ft, we see that the width x = 35 ft. The area is the product of these measures, so
A = 525 ft². Let’s add these last two number experiments to our table.

At this point, the last set of dimensions yields the maximum area, but is it possible
that another choice of x and y will yield a larger area? Experiment further with numbers
of your choice to see if you can find dimensions that will yield an area larger than the
current maximum in the table, namely 525 ft².

Let’s now call on what we’ve learned in this section to attack this model. First,
we’re constrained by the amount of material we have for the job, a total of 80 ft of
fencing. This constraint requires that 3 times the height of the garden, added to the
width of the garden, should equal the available amount of fencing material. In symbols,

x + 3y = 80. (18)

We’re asked to maximize the area, so we focus our efforts on finding a formula for the
area of the rectangular garden. Because the area A of the rectangular garden is the
product of the width and the height,

A = xy. (19)

We now have a formula for the area of the rectangular garden, but unfortunately we
have the area A as a function of two variables. We need to eliminate one or the other
of these variables. This is easily done by solving equation (18) for x.

x = 80 − 3y (20)

Next, substitute this result in equation (19) to get

A = (80 − 3y)y,

or, equivalently ,

A = −3y² + 80y. (21)

Note that we have expressed the area A as a function of a single variable y. Also, the
function defined by equation (21) is quadratic. Compare A = −3y² + 80y with the
general form A = ay² + by + c and note that a = −3 and b = 80 (we have no need
of the fact that c = 0). Therefore, if we plot A versus y, the graph is a parabola that
opens downward (see Figure 6), so the maximum value of A will occur at the vertex.
The y-coordinate of the vertex is found with

To find the width of the rectangular garden, substitute y = 40/3 into equation (20)
and solve for x.

Thus, the width of the rectangular garden is 40 ft. We can find the area of the garden
by multiplying the width and the height.

Note that the resulting area, is only slightly bigger the last tabular entry
found with our numerical experiments .

You can also find the area of the rectangular region by substituting y = 40/3 into
equation (21). We’ll leave it to our readers to check that this provides the same
measure for the area. You will also notice that the second coordinate of the vertex in
Figure 6 is the maximum area A = 1600/3 ft².

Figure 6. The maximum area, A =
1600/3 ft², occurs at the vertex of the
parabola, (40/3, 1600/3).