### Lecture

Optimization^{Problems}

Now

### that

^{we}know the

^{method}to

### find

^{the}

^{absolute}

max min

### of

^{a}

^{function}

^{we}

^{can}

^{use}

^{this}

^{to}

### solve real world problems

### 8 g ^{A} farmer ^{has}

^{800M}

### of fencing ^{and}

^{wants to}

### fence

^{a}

### field _{Assuming}

^{one}

^{side}

### of

^{the}

### field ^{is}

^{a}

^{aimer} ^{find}

^{the}

^{dimensions} of

the

### rectangular field

^{that}

_{gives}

^{the}

_{largest}

area

### Sol

ew w The

### situation of

^{the}

problem

### is

as shown in### _River

the### figure

### The

^{Area}

### l

wo The

### farmer

^{has}

^{800}

^{m}

### of ^{fencing} ^{and}

^{one}

^{side}

### is never

p

### l

12W 800### D l

800 2W8

### Area

^{800}

^{2W}

^{w}

^{800W}

^{2W}

o we want to

### maximize

^{the}

^{area}

^{p}

^{we}

^{want}

### to find

^{w}

^{s't}

^{Area is}

^{maximum}

### Clearly

^{wzo}

^{Now}

^{o}

^{l}

^{12W}

^{800}

^{the}

maximum amount ^{we} ^{could be} is 400 _{p WE} 0,4003

We

### have

### Area

800 4W^{o}

^{w}200

ooo w 200

### is

the critical### point

We ^{now} _{compute}

### A lo

OA 200 800 200 2 20072

160000 ^{2} ^{40000} ^{80000}
A ^{400} 800 400 2 ^{40032} _{0}

ooo

### the

maximum### Area

^{is}80,000 m

Qued Suppose ^{we}

### have

^{300cm}

### of ^{tin}

^{to}

^{work with}

### and

^{we}want

### to

make the_{biggest}most awesome

### soup

^{can}

^{ever}

^{How}

^{much}

^{soap} ^{could}

^{our can}

^{hold}

### Sot

### y f

^{The}

^{in}

^{situation}

^{the}

^{diagram}

^{8 shown}

We want to maximize the volume

IT

### 82k

The amount

### of

tri available^{300cm}

D

### jTr2

^{sty}

^{1}

^{211Th}

^{300}

her

### top

^{bottom}

^{side}

21182 21184 300 D

### h

^{300}

^{211}

### 82

211^{8}

150 8

D vote ^{IT}

### 8211,5

^{g}

^{1508}

### iT83

We _{want to} _{maximize} _{vol} So we

### need bounds

on 8

### clearly

^{rzo}

Also o we

### have

300cm### of

^{tin}

^{we}

^{must}

^{have}

211

### 82

_{E}300 P

### TZE

_{1502}

ST

RE

JT

### Io

^{0182}

### 151

_{IT}

o ol 150 ^{311}

### 82

0D 11

### 82

^{50}

^{D}

^{8}

### 150

### So

^{we}compute

### Vollo

^{O}

### Volffisft

^{o}