CalcAB
Chp 44A
Optimization Models
Notes
Students, on your own papers you will be writing the actual calculation work
and answers to each problem.
The notes here are intended to guide how/why to address different kinds of problems.
1.
OPTIMIZATION problems ask you to solve for a measurement or quantity that will MAXIMIZE an
amount such as volume, area, or profit.
Or, MINIMIZE an amount such as cost, distance, or
perimeter.
The problems will mention (or imply) something is to be "maximized" or "minimized".
We will be
working with
extreme value points
; the
maximum and minimum turning points
studied in
Chp 41 and Chp 43.
The xcoordinates of extreme value points are called
critical numbers.
They are calculated by setting a first derivative equal to 0
.
(Application problems are
real
world
type of situations; it is not appropriate to consider the derivative as undefined.)
2.
Ex(volume of an open top box)—
An open top box is to be made from a square piece of cardboard, 8 inches
on a side, by cutting equal squares from each corner and turning up the
sides (see figure).
(a) Write the volume as a function of x.
(b) What dimensions of the box will hold maximum volume?
(c) What is the maximum volume?
(a) Equation for volume
Need the geometry formula for volume
of a rectangular prism.
V = lwh
Write algebraic expressions to use for the sides of the prism
.
To turn a rectangular piece of cardboard into a box the four square corners need to
be cut out; all with the same size side x.
Then, fold up the four flaps to create an
opentop box (rectangular prism).
The length of the original cardboard is 8 inches when the squares are cut out from
each end.
That means 2x is cut off each side of the original cardboard.
Need
algebraic expressions for the length, width, and height of the prism formed.
(b) Dimensions of the box with largest volume.
Unit of measure for length is inches
.
Need to solve for x
;
x is the square cut out of the original 8inch cardboard.
This problem is only interested in the box with the largest volume.
In Part(a) the volume equation developed
represents the volumes of all possible boxes.
Look at the graph of V(x).
It has two turning points.
(You will need to
change the window setting to show yvalues up to 45.)
The turning point that is a maximum is the point needed
.
The xcoordinate can be used to get the dimensions of the largest box
and the ycoordinate is the maximum volume
.
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 Spring '10
 KNOPF
 Rectangle, product equation

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