RECITATION 11: 4-7 OPTIMIZATION (PART 2)

STARTER PROBLEM: Write out the steps (hoops) you will need get through in order to produce a com-plete and correct answer to an optimization problem.

PRACTICE PROBLEMS:

1. Find the area of the largest rectangle that can be inscribed in a semicircle of radius 10.

Uses a calculator 1 Recitation 11

;Determine the dutyto be maximized or minimizedWrite that quaggasa function of one variable and determine

.

rItnsImIiIipts.uss@caosedgnonrrmualnoratIIu.rorZrdder ) to identify where the man

• Answer the question

lo@y_yofytlooFwewanttomaxim_zeauaA-2x.y= Zx FE where [ 0 , ID is thedomain

Closed - Interval test- ×

A' G) = 2. 1 . ( no

.EE#.'zGoo.x2Y2c2x)//o0$5E

A 2 . 5fz . Do

=Zoo -2×2-2×2 = Zoo⇒F. ' TheTeargasntghulanrasesstanpoasi"= 4150¥TE 200 units squared .

critical pt : x= Go = SE,

X= 10

2. A retailer has been selling 1200 tablet computers a week for $350 each. The marketing departmentestimates that an additional 80 tablets will sell each week for every $ 10 that the price is lowered.

(a) Find the demand function.

(b) What should the price be set at in order to maximize revenue?

(c) If the retailer’s weekly cost function is C(x) = 35, 000 + 120x what price should it choose inorder to maximize its profit.

Uses a calculator 2 Recitation 11

p = pricep - 350=-18 ( q - 1200 ) .

q= # tablets sold-

pts : ( 1200,350 ) |p=- tqqt 500( 1280,340 ) _|340-350 - 10

me '¥ = Teo'

Too" to

Dq 0

RL g) =p .q=( to 9+500)q=tgq2 +500g , domain :( 0 , -)

Answer:

R " g) = - 14 q +500=0 -The price

that will maximize

of= 2000 revenue is

R "t 14

3. An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline isto be constructed from the refinery to storage tanks located on the south bank of the river 10 kmdownstream of the refinery. The cost of laying pipe is $ 10,000/ km over land to a point P on theopposite bank and then $ 40,000/km under the river to the tanks. To minimize the cost of pipeline,where should P be located?

4. Two posts, on 12 feet high and the other 28 feet high stand 30 feet apart. They are to be stayed bytwo wires, attached to a single stake, running from ground level to the top of each post. Whereshould the stake be placed to use the least amount of wire?

Uses a calculator 3 Recitation 11

Cost ( in 10,000 's )Refinery P

•=*=•f4H€c(×)= 1. ( io - x ) + 4. ( 4+5/2

2K¥ f•2¥ where x can take values [ 0,10 ] .F-'teams C 'lx)= - 1+4 .tz( 4+5542×1=46 - I10 km 4txTcritical points : 4×= (4+E)

"2

or 16×2=4*2 . Thus,

E- 445.

So ×=%fs .

×¥Fj¥h!m¥¥t45mn Apttshondbe located to - To km downriver from the refinery .

pg.IM4×7=(122*442+430*+282)

"

; domain :[ 0,3g

L' ( *)=¥(144+1-254262×7#30*5+282%2430. DHD

× 30- ×

tEE@oETo2.o+ + + ← sign of "

critical pts : 11-1×(z(o-42+28442=(30*7444+5)

"20 q 30

×2( goo -60×+1-2+7847=940-60×+13) (144*2) And :

X4 - 60×3+1684×2=114-60×3+1044×2 - 8640×+129600Length Is minimized if

- --

-the wire is fixed

640×2+8640×-129600=0 9ft from the 12ft post .2×2+27×-405=0a) ÷ 320x=q

5. Four feet of wire is used to form a square and a circle. How much of the wire should be used forthe square and how much should be used for the circle to enclose the maximum area?

Uses a calculator 4 Recitation 11

.

✓4ft wire

area enclose = Alex )= area in + area incircle square

4- x 1

/ \,

=

#x2t ,I( 4- x5 ; domain :[ 0,4 ] .

GsqNow

,A' G)

=gI+ too ( 4-

×)tD=€-

4¥=o)•8tcircumference

1 perimeter 's 4* Criptniscal : 4×-4 # tax to or ×= 44,45=1.76isx

'

,- - - o + + + ← Sign A '

So :1 So : I 1 II

×=2tr or , 4- ×=4s or 0 1.76 4

r= I 1 2 location oflocal Minn .

ZH'

l,

S= 4¥ .,

X=o)ACo)=lj x=4,

A (4) =4z > 1 .Thus 2,

Thus,

adf.ge#=T' (Ea) i area So , area is maximized when all of' square

= ,I( 4*122 the wire is used to make the circle

X1

= - i

41T ,