Transcript Optimization

Optimization
Objective
 To
solve applications of optimization
problems
 TS: Making decisions after reflection
and review
Optimization
 Optimization
is the procedure used
to make a design as effective as
possible.
Number Problem

The product of two positive numbers is
288. Minimize the sum of twice the first
number plus the second number.
x  first number
y  sec ond number
sum  2 x  y
x y  288
Primary Equation
Secondary Equation
x  first number
y  sec ond number
Number Problem
sum  2 x  y
Primary Equation
x y  288
Solve for y
Substitute y into
Primary Equation
Simplify
Differentiate
Secondary Equation
288
y
x
s  2x 
288
x
s  2 x  288 x 1
ds
 2  288 x 2
dx
ds
 2  288 x 2
dx
0  2
Find critical points
2
288
x2
288
x2
2 x 2  288
x 2  144
x  12
and x  0
0
und
0
Test
12
0
12
Min
dA
dw
Number Problem
x  12
288
y
x
288
y
12
y  24
Numbers:
12 & 24
The Fence Problem

A farmer has 100 ft of fencing to enclose a
rectangular field. The field will have one side
along his farmhouse, and thus needs to be
fenced on only three sides. What are the
dimensions of the rectangle that will maximize
the area?
Maybe it looks like this…
or this…
or this.
The Fence Problem

A farmer has 100 ft of fencing to enclose a
rectangular field. The field will have one side
along his farmhouse, and thus needs to be
fenced on only three sides. What are the
dimensions of the rectangle that will maximize
the area?
l
A  lw
w
w
l  2 w  100
The Fence Problem
A  lw
Primary Equation
l  2 w  100
Solve for l
Substitute l into
Primary Equation
Simplify
Differentiate
Secondary Equation
l  100  2 w
A  (100  2w) w
A  100w  2w
dA
dw
 100  4w
2
The Fence Problem
dA
dw
Find critical points
 100  4w
0  100  4w
4w  100
w  25
0
dA
dw
Test
25
MAX
The Fence Problem
w  25
l  100  2 w
l  100  2(25)
l  50
Dimensions:
25 feet  50 feet
The Box Problem

An open box is to be constructed from a piece of
cardboard which is 16” by 13”, by cutting out a
square from each of the four corners and bending
up the sides. What size square should be
removed from each corner in order to create a
box that maximizes volume?
The Box Problem

An open box is to be constructed from a piece of
cardboard which is 16” by 13”, by cutting out a
square from each of the four corners and bending
up the sides. What size square should be
removed from each corner in order to create a
box that maximizes volume?
The Box Problem
V  l  wh
w
h
l
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l  16  2h
w  13  2h
h
h
The Box Problem
V  l  wh
Substitute l and w
into Primary Equation
Simplify
l  16  2h
w  13  2h
Primary Equation
Secondary Equations
V  (16  2h)(13  2h)h
V  (208  32h  26h  4h )h
2
V  208h  58h  4h
2
Differentiate
dV
dh
3
 12h  116h  208
2
The Box Problem
dV
dh
 12h 2  116h  208
Find critical points
0  12h  116h  208
Quadratic Formula
h  7.288 h  2.378
2
0
0
dV
dh
Test
2.4
7.3
MAX
MIN
Dimensions of square:
2.4"  2.4 "