_River of of

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

e

w w The

situation of

^the

problem

is

as shown in

_River

the

figure

The

^Area

l

w

o The

farmer

^{has 800 m}

of ^{fencing and}

^one

^side

is never

p

l

12W 800

D l

800 2W

8

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

O

A 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

^{Thein situationthe}

^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

211⁸

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

0

D 11

82

^{50 D 8}

150

So

^we compute

Vollo

^O

Volffisft

^o

vous tH 4