### 4.4 Modeling and Optimization

``` Reminder:
The Extreme Value Theorem states that every continuous
function on a closed interval has both a maximum and a
minimum value on that interval.
(The max and min MAY occur at the endpoints.)
 An 11 by 8.5 inch piece of cardboard is to be made into a
rectangular open-topped box by cutting out a square at
each corner. What dimensions will produce a box with
the maximum volume?
x
x
x
x
x
x
x
x
 An 11 by 8.5 inch piece of cardboard is to be made into a
rectangular open-topped box by cutting out a square at
each corner. What dimensions will produce a box with
the maximum volume?
Length of box = 11 – 2x
Width = 8.5 – 2x
Volume = x(11 – 2x)(8.5 – 2x)
Volume = 93.5x – 39x2 + 4x3
V’= 93.5 – 78x + 12x2= 0
x = 4.915
x = 1.585
(critical points)
Height = x
Now what???
Finding max, so take derivative
 An 11 by 8.5 inch piece of cardboard is to be made into a
rectangular open-topped box by cutting out a square at
each corner. What dimensions will produce a box with
the maximum volume?
V’= 93.5 – 78x + 12x2= 0
x = 4.915
x = 1.585
To find max/min critical point, use the First Derivative Test.
Is there a restricted interval I must keep my “testing”
numbers in between? 0 ≤ x ≤ 4.25
f’(1)
f’(2)
Inc. Dec.
0 1.585
4.25
MAX
 An 11 by 8.5 inch piece of cardboard is to be made into a
rectangular open-topped box by cutting out a square at
each corner. What dimensions will produce a box with
the maximum volume?
Since x = 1.585 produces a max, we need to cut
out a 1.585 by 1.585 square from each corner.
Length of box = 7.83 in.
Volume = 66.148 in3
Width = 5.33
Height = 1.585
 Identify all given quantities and quantities to be
determined.
 Write an equation for the quantity to be maximized or
minimized.
 Be sure the equation is reduced to have only one
independent variable.
 Take the derivative and find critical points.
 Determine the feasible domain of the equation so you
will know which of the critical points is/are important.
 Use First Derivative Test to find absolute max/min.
 BE SURE TO CHECK YOUR ENDPOINTS…many
optimization problem solutions occur at the
endpoints.
 Which points on the graph of y = 4 – x2 are closest to the
point (0, 2)?
 A rectangular page is to contain 24 inches of print. The
margins at the top and bottom of the page are to be 1.5
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?
 Four feet of wire is to be used to form a square and a
circle. How much of the wire should be used for the
square and how much should be used for the circle to
enclose the maximum total area?
 A utilities company wants to deliver gas from a source S
to a plant P located across a straight river 3 miles wide,
then downstream 5 miles. It costs \$4 per foot to lay the
pipe in the river but only \$2 per foot to lay it on land.
How can the pipe be laid most economically?
```