2 met

DESIGNING A WATER TANK

A square metal sheet (2 metres by 2 metres) is to be made into an open-toppedwater tank by cutting squares from the four corners of the sheet, and bending thefour remaining rectangular pieces up. to form the sides of the tank. These edgeswill then be welded together.

* How will the final volume of the tank depend upon the size of the squares cutfrom the corners?

Describe your answer by:

a) Sketching a rough graph

b) Explaining the shape of your graph in words

c) Trying to find an algebraic formula

* How large should the four corners he cut.

tank is as large as possible?%O that the resulting volume of the

Shell Centre for Mathematical Education, University of Nottingham. 1985.

2 metres

Ic ~‘I

DESIGNING A WATER TANK... SOME HINTS

Imagine cutting very small squares from the corners of the metal sheet. Inyour mind. fold the sheet up. Will the resulting volume be large or small?Why?

Now imagine cutting out larger and larger squares.

What are the largest squares you can cut? What will the resulting volume be?

Sketch a rough graph to describe sour thoughts and explain it fully in wordsuntie r neat h:

Volume ofthetank (m’)

Length of the sides of the squares (m).

* In order to find a formula, imagine cutting a square x metres by x metres

from each corner of the sheet. Find an expression for the resulting volume.

* Now try plotting an accurate graph.

(A suitable scale is 1 cm represents 0.1 metres on the horizontal axis, and1 cm represents 0.1 cubic metres on the vertical axis).

How good was your sketch?

* Use your graph to find out how large the four corner squares should be cut,

so that the resulting volume is maximised.

©Shell Centre for Mathematical Education, University of Nottingham, 1985.

THE POINT 01” NO RETURN

Imagine that you are the pilot of the light aireralt in the picture. which is capableof cruising at a stead~ speed of 3(M) km h in still air. N ou hate enough fuel onhoard to last four hours.

‘You take oil Irom the airtield and. on the out~~ard journe\ . are helped along h~aSS km h ~ md ~ hich increases sour cruising speed relati~e to the ground to 350km h.

Suddenly you realise that on sour return ourne~ ~ou ~ ill he flying into the ~ mdand ~ ill therefore slots down to 250 km h.

What is the maximum distance that ~ou can travel from the airfield, and stillhe sure that ~ou hate enough fuel left to make a safe return journeY?

In’. estigate these ‘pornts of no return’ br different wind speeds.

©SheIl Centre for Mathematical Education, University of Nottingham, 1985.A’~7 ‘ICfl’

THE POINT OF NO RETURN... SOME HINTS

* Draw a graph to show how your distance from the airfield will vary with time.How can you show an outward speed of 350 km h?How can you show a return speed of 250 km h?

800-

700—

~ 600- ::::

400-

S2 300— : : : : : : : : : :

U200- : : ::::. : :::: :::::

1(X)-

0 [ 2 3 4Time (hours)

Use your graph to find the maximum distance you can travel from theairheld. and the time at which you should turn round.

On the same graph. investigaw the ~points of no return’ for different windspeeds. What kind of pattern do these points make on the graph paper?Can you explain why?

Suppose the windspeed is w km/h.the point of no return’ is ci km from the airfield

and the time at which you should turn round is thours.

Write down two expressions br the outward speed of the aircraft.one involving wand one involving d and!.

Write down two expressions for the homeward speed of the aircraft,one involving wand one involvingdand t.

Try to express d in terms of only t. by eliminating it’ from the two resultingequations.

Does this explain the pattern made by your points of no return’?

©Shell Centre for Mathematical Education, University of Nottingham, 1985.AQ (lcfl

“WARMSNUG DOUBLE GLAZING”

(The ~ indows on this sheet are alldrawn to scale: I cm represents I loot).

How have “Warmsnug” arrived atthe prices shown on these windows?

Which window has been givenan incorrect price? How muchshould it cost?

©SheIl Centre for Mathematical Education, University of Nottingham, 1985.

Explain your reasoning clearly.

49 (154’

“WARMSNUG” DOUBLE GLAZING... SOME HINTS

* Write down a list of factors which may affect the price that “Warmsnug” ask

for any particular window:

e.g. Perimeter,Area of glass needed,

* Using your list, examine the pictures of the ~‘indows in a systematic manner.

* Draw up a table, showing all the data which you think may be relevant.

(Can you share this work out among other members of your group?)

* Which factors or combinations of factors is the most important in

determining the price?

Draw scattergraphs to test your ideas. For example, if you think that theperimeter is the most important factor, you could draw a graph showing:

Costofwindow

Perimeter of window

* Does your graph confirm your ideas? If not, you may have to look at some

other factors.

* Try to find a point which does not follow the general trend on your graph.

Has this window been incorrectly priced?

* Try to find a formula which fits your graph, and which can be used to predict

the price of any window from its dimensions.

CShelI Centre for Mathematical Education, University of Nottingham, 1985.Sn (1SS~

PRODUCING A MAGAZINE

A group of bored, penniless teenagers want to make some mone~ hy producingand selling their own home-made magazine. A sympathetic teacher offers tosuppl~ duplicating facilities and paper free of charge. at least for the first fe~issues

I a) Make a list of all the important decisions they must make.Here are three to start you oft

1’Iz

-c

zz

-C

2 The group eventually decides to find out how many potential customersthere are within the school, by producing a sample magazine and conductinga survey of 100 pupils. asking them “Up to how much would you be preparedto pay for this magazine?” Their data is shown below:

Selling price (s pence) Nothing 10 20 30 40

[~Nmb~ prepared to pay this price (n people) 100 82 58 30 18

How much should they sell the magazine for in order to maximise theirprofit?

3 After a few issues, the teacher decides that he will have to charge the pupilslOp per magazine for paper and duplicating

How much should they sell the magazine for now?

©Shell Centre for Mathematical Education, University of Nottingham, 1985.

How long should the magazine he? (1 pages)

How many writers will be needed’! (w writers)

How long will it take to write? (r hours,

h) Many items in your list will depend on other items.For example,

For a fixed number of people involved.the longer the magazine, the longerit will take to write. _____________

I pagesFor a fixed length of magazine.

the more writers there are, .

Complete the statement, and sketch a graphto illustrate it.

Write down other relationships you can find.and sketch graphs in each case.

w writers

51(158)

PRODUCING A MAGAZINE... SOME HINTS

Here is a more complete list of the important factors that must be taken intoaccount:

Who is the magazine for? (schoolfriends?)What should it be about? (news, sport. puzzles. jokes.How long should it be? (1 pages)How many writers will it need? (w writers)How long will it take to write’? (t hours)How many people will buy it? (ii people)What should we fix the sellingprice at? (s pence)How much profit will we makealtogether? (p pence)How much should we spend onadvertising? (a pence)

* Can you think of any important factors that are still missing?

* Sketch graphs to show how: tdepends on w; w depends on 1;

n depends on s; p depends on s: n depends on a.* Explain the shape of each of your graphs in words.

2 * Draw a graph of the information given in the table of data.* Explain the shape of the graph.

* What kind of relationship is this?

(Can you find an approximate formula which relates n to s?)* From this data, draw up a table of values and a graph to show how the

profit (p pence) depends on the selling price (s pence).(Can you find a formula which relates p and s?)

* Use your graph to find the selling price which maximises the profit made.

3 Each magazine costs lOp to produce.* Suppose we fix the selling price at 20p.

How many people will buy the magazine? How much money will beraised by selling the magazine, (the tevenue’)? How much will thesemagazines cost to produce? How much actual profit will therefore bemade?

* Draw up a table of data which shows how the revenue, production costs

and profit all vary with the selling price of the magazine.* Draw a graph from your table and use it to decide on the best selling price

for the magazine.

©Shell Centre for Mathematical Education, University of Nottingham, 1985.52(159)

There are 6 main stations along the 13’ 2 mile track:

(The distances between them are shown in miles)

Tan-v-Bw~ch

Port h~~og ~&~~i

Three steam trains are to operate a shuttle service. This means that they willtravel back and forth along the line From Porthmadog to Blaenau Ffestiniog witha ID-minute stop at each end. (This should provide enough time for drivers tochange etc.)

The three trains must start and finish each day at Porthmadog.

The line is single-track. This means that trains cannot pass each other, except atspecially designed passing places. (You will need to say where these will beneeded. You should try to use as few passing places as possible.)

Trains should depart from stations at regular intervals if possible.

The journey from Porthmadog to Blaenau Ffestiniog is 65 minutes (includingstops at intermediate stations. These stops are very short and may be neglected inthe timetabling).

The first train of the day will leave Porthmadog at 9.00 a.m.

The last train must return to Porthmadog by 5.00 p.m. (These times are morerestricted than Those that do. in fact, operate.)

aTHE FFESTINIOG RAILWAY

This railway line is one of the most famous in Wales.Your task will be to devise a workable timetable forrunning this line during the peak tourist season.

The follo~ ing facts will need to be taken into account:— ‘4

Ta,IsIa~_/rN.%, Iae n a u

Ffestiniog

Minffordd

©Shell Centre for Mathematical Education, University of Nottingham, 1985.5~ (164~

THE “FFESTINIOG RAILWAY” ... SOME HINTS

Use a copy of the graph paper provided to draw a distance-time graph tor the9.00 am. train leaving Porthmadog.

Try to show, accurately:

• The outward journey from Porthmadog to Blaenau Ffestiniog.

• The waiting time at Blaenau Efestiniog.• The return journey from Blaenau Fiestiniog to Porthmadog.• The waiting time at Porthmadog . . . and so on.

What is the interval between departure times from Porthmadog tiw the abovetrain?How can we space the two other trains regularly between these departure times?Draw similar graphs for the other two trains.How many passing places are needed? Where will these have to he?From your graph. complete the following timetable:

Station Daily Timetable

Porthmadog d 09(k)

Minffordd d

Penrhyn d

Tan-y-Bwlch d

Tanygrisiau d

Blaenau Flestiniog a

Blaenau Flestiniog ci

Tanygrisiau d

Tan-y-Bwlch d

Penrhyn ci

Minffordd d

Porthmadog a

Ask your teacher for a copy ofcompares with your own.

the real timetable, and write about how it

©SheIl Centre for Mathematical Education, University of Nottingham, 1985.54(165)

4

(sainu) ~opewquo~ wosj aounls!Q

©Shell Centre for Mathematical Education, University of Nottingham, 1985.55 (166)

Take the lamous Ffesliniog Railway for a memorable journey through theSnowdonia National Park. From the coast at Porthmadog the little train

p climbs through tranquil pastures and magnificent forests, past lakes andwaterfalls round horseshoe bends and even a complete spiral, sometimesclinging to the side of the mountain and sometimes tun~elling under it.Much of the area is so remote that there are not even x.iy motor roads andthe train stops occasionally at isolatedcottagesve.ose Inhabitantsdapendentirely on the railway 13i4 miles and o’., hours journey time fromPorthmadog is Blaenau Ftestiniog. over 7)0 feet above sea level, Here arethe slate mines at Ltechwedd and Gloddfa Ganol which are both open tovisilors

To cater for all your requirements there are gilt shops at Porthmadog.— Tan-y’Bwlch and Biaenau Ffestiniog a sell-service restaurant at

Porthmadog and station buffets at Tan-y-Bwlch and Bfaenau FfestiniogSo sit back, relax and take the journey of a lifetime Let our stewards wait

— on you with snacksand drinksfrom thebufletcarorminibarfrolley For theenthusiast, there’s even more many of the trains are pulled by uniqueand historic steam locomotives some of which have served the line for overa hundred years

Your complete day out...SPECIAL INCLUSIVE EXCURSIONS

FROM PORTHMADOGSTWLAN DAM (Daily 26 May to 13 September)Depart Porfhmadog by mosi “daytime” trains (see ilmetablel for busconnection from Tanygrisiau This is one of the most spectacular ‘busroutes in Britain. ascending 10 a height of 1650 teei The bus returns 10Tanygrisiau station trom where you may return to Porthmadog or. 1 you sowish. conlinue your rail journey to Blaenau Flesiiniog at no extra chargeAllow 2i1, hours for the complete excursion 1 returning direci Iroml’anygrisiau or 3 hours 20 minuies if retorning via Blaenau Ffesiiniog

LLECHWEDD CAVERNS (Monday to Saturday 30 March to 2 November.also Sundays 26 May lo 6 September)Depart Porthmadog by any train up to 1220 1310 when operatrngfTransfer to bus at Blaenau Flestiniog for shori trip across iown tu

— Llechwecld Caverns Then take either the battery electric train or ihe den,nine incline into theheart of a Victorian slafe mine Ailowal least S hoursiu

do lustice 10 the cornpleie excursion

FFESTINIOG LINK TOURS (Monday 10 Friday 27 May10 13 Sepiember)The new point Flestirriog Btitish Rail station in Blaenau Flestinioq hasenabled us to provide easy rail accessto the Conwy Valley arid North WalesCoast Inc journey Irom Porihmadog 1° IlanoLidno oilers 44 miles 01$pecraciji;Ir niouritriin rind coastal scenery Depart Portrirnanog cr0950 for0540 when opetatingf for itlis highlight ol yourhoiiday A shorier veisiori oftn.s Ic Ut trot); Pot tiiniadog 10 Betws’y-Coed oi Llaniwsi ‘a also available

GLODDFA GANOL Slate Mine. Free admission will oe ‘jrariieddur’i.q lit.’ 19S5 season try any child whose patent produces killtclt.rrr Fiostti,i ug tickt’i bt’tweoti Porlhr.iadog and Bidotiami

cxli 109 A bus soviet; ope rite S heiween Bi;iena., F test iii iOtlsat ni, mu 3 o iris (“i to;

~a ~r’i~t g~~ai .6~eeic ‘tzains ct (Vain3, iV4NI) “t It Ir(’yj

t_i,lE,’el;i ‘ii I I. ‘si/s. it iii tt:iyl’ I, i wry it iht, ta iti.ruittin:r,Srirt’iC., ~‘sir.:~’,tY r .is...LVN ru5 L5’IAY ‘AlL (I’ tfflt’if)istt,Ali.VmAY if’iit~%t ‘5’ ~•/‘%/ WL1. ‘i~OO’ ‘5NL I SNI Still i(unii

A’? ,“ai’r ~l r lei . ‘MI Se ‘Vt I )f ft fIGi I. ANt) PAIL WAII iCrlrl F’i’fi S ‘i’! S

‘ti’’ l’”r,,,1eJ ‘ann jul.’’ iF 76 ‘ri

MORE MILES FOR YOUR MONEYrHiRD CLASS ir mt yaw a,aiiabi eat

Suppiemenraty cr.argef

* ti’a,reiout by drebei seituci’ Showit Discs on tumsiabit, tSer~itit fly ANY ti’’

Reductions for Children and Senior Citizens as follows;Children under S free ONE CHILD UNDER IS TRAVELS FREE IN THIRDCLASS FOR EACH ADULT PAYING THIRD CLASS ORDINARY ORECONOMY FARES Additional children aged Sand under 16 travel ai traitfareSenior Citizens travel at halt fare on return lares only

PLEASE NOTE FREE CHILD FACILITYFamily Fares up to 22% cheaper than three years ago!Fares co.,ectai i.e oiyu..rq t,t,’.ussi,r,tI.,u:uh’tuaii,’, . it,,uui

Drd you knowThat the Flestrosoy Railway lids a supporters 011Th

The FESTINIOG RAILWAY SOCIETYis a voIunlaryorganisation deutcated tostipportirig lb I d

existence of the Flestiniog Railway

You caii1ouriaionr,ui me Rauiw.iy Satiops mit s,’i OLfrfli iuitho;iqeot SI to tie Kloititit’rstiil) S.’trt’i;ruv J t.i,iti ty ‘1KWiumctii;stei Hauls 502.1 901) IM.’ntl Cr’ ‘‘,‘ice’ int’l Iii

I y i ti;iiJa hut’

if yuti would lilt’ hurt iii ri ifuiirr.i t it ibm 111,1 in I • ri, lw.Iy• ‘iiSuctc’t’,’,tsk ‘di its.’ bookitiy cnits iii I ily r ihi’t,it it .4 ii U.irs its.’ Fesri,I,og Railway Sr,cioiy

Ffestiniog Railway. Porlhmadog. GwyneddTelephone No’ (0766) 2340/2364

Meitibet 1 ott T A I Noult Wt

S ,~

ci

• _). - .- —

“‘. “~1

tiestinloRhe;I(fo,dd ~_ Railway Rheilf toyed ~, Railway

MOUNTAINS, LAKES AND COASTLINE

festinloRheilttordd Railway Rheilltordd Rairway

Rtre,ttto’dd Railway Rheilttordd Railway

Porthmadog toBiaenau Ffestiniog

Time Table 1985

Pruncupal Fareslavaulabil, in vutrier direclioii

Porihmadog to Biaenau FteautniogPorthmadog to Tan’t’ OwlchT.n’y’ Bwicit 10 Bl.enau FlestlniogPorthmadog to Penrhyn

CHILDREN FREE

ORDINARY ORDINARY ECONOMY*SINGLE RETuRN RItuRN

£2.00 £5.50 £4.50£1.70 £3.40£1.70 £3.40lop £1.00

J

One fta4czea9err (ada ~6)Czaucü jza jet each adalt

ftae~ au Std e&zaa jaa

ire Sitter‘II’ arid

P,;i,i,v.t,,’tlit’d ii,tuii’,ii;v i Sri)’ .4.

HIGHSUMMEN

0840Porthmadog d.

Monday 15 July to FrIday 30 August(Also Spring HolIday Week Sunday 26 May to Thursday 30 May)

PwOln90 0 0753

~ZD -3 ~lei.Oro,08 I 0909

0950 1035 1125 1220 1310MONDAYS TO THURSDAYS

1400 1455 1545

i:.’ 0959- 1005

• 10251043

I’’’ 1055

Mlnftordd d.Penrhyn d.Tan-y-Bwlch d.Tanygrisiau d.SI. Ffestiniog a.

elFt..?0&.g 0- “ L0II?0~dv0 JO~

- L I’OkOOO I— andudac 4

Llsndi,0,0 SC, I97 70.09.159

00551040

1341140120512231233

1635 ~it

1640770

‘9350210

12291235125513131325

0950 II’$IS 1125

‘55217

1319132513451407

FRIDAYS, SATURDAYS AND SUNDAYS

07559 09390426e 059

400 7000433 530

95430422

1504 15541510 16001530 16201548 1641

1409141514351458

0945042

0055043008400940

0530004

10441050111211351146

11551201122512451250

“35

051204‘2110950‘006~ 00

1220 11(11111400 1455 1635400 bOO

1417 1507 1600 1652 1740330 4501019540‘44l 7550

1101121224

SI. FlestiniogTanygrisiauTan-y-BwlchPenrhynMlntlordd

d.d.d.d.d.

0677 0955 - 7755,, 0247.0 343090? 7040 ‘137,, 1325s4 412

0959 I’’ 1134 1229 1409 15041005 t I 1140 1235 1415 15101025 1205 1255 1435 15301043 1223 1313 ‘ 1458 15481055 1233 1325 1507 1600

615771071,16

7132 7450343 ,00l

‘445 009

I 1105I 1111•‘ 1133

I 11491154

7205246

203

12451251131513351340

90209409

16441650171017281740

1404 ‘434

1644 ‘i’

165017101728

0746 2020‘445 2?!?lt03 2117630 jul640 003

‘74? 903

1752 I

1814 I’

1830 I

1835 a’

2332~2239

13401346140514241429

105,0200’o

02l4’~ 0950009407000’09400,00.

40,60,00 09i,,.outh04i,tl0,00 Up—Os—

14301436150015201525

15201526155016061611

0210‘7,,

1621164016591706

1330 045043? ‘546

1441 556 -~jOo — ISis7??? ‘343224 445

1706173017461751

Porthmadog a. 1209

I

l625 ‘7491724 9445‘734 990‘450 7930170? 7647609 ‘76?

1615 1700 1746

‘433 530574 676

71279450

1304 1354 1444 1539 1629 1719 1805 1848~

Saturday 31 August to Sunday 15 September

T600~0533

1105 1245 1340 ‘N’115201111 I 1251 1346 +1115261133 1315 1405 jijiliSso1149 1335 1424 tt116061154 I 1340 1429 t.441611

‘9552036

79341901

16151621164016591706

17461752181418301835

MONDAYS TO FRIDAYS

205 ‘43375,4

713 70 3255 4117303’ 73535 1450

1209 IICIl~I1354

DAILY

Porthmadog d.0950l11fl11125 1220 ll’11111400 1455 1635 0950 1220 1455 0950 1220

06551040

MinttorddPenrhynTan-y-BwlchTanygrisiauSI. Ftestiniog

Monday 16 Sept.SATS. & SUNS. to Sunday 3 Nov.

‘ PwIlhst 4 0755 0938 1135 500 0434 ‘‘35 o,o4 0 SSq 7?‘ ‘ 4 6416000,00 4 0915 004 205 530 0977 1205 7433 092 205

.. -. 4961106041 4. 0977 343 520 047? 0655 1343 09?? 0955‘ ~!~9≤~ 4., 0909 422 606 090? 040 7412 0907 1040

530 9790 ‘955616 ‘759 2036

0608— — ‘434 52727633 940? 2130

144411*111625 1719 1848

Oaily26 Dec 10

I JanSats & Suns

15 Feb 1966 to23 March 116$

d. 0959 I’’ 11341229 fl1409d. 1005 I ‘11401235 +11415d. 1025 1205 1255 ‘M1435d. 1043 12231313 ‘1)41458a. 1055 ‘a 1233 1325 fr11507

15041510153015481600

60 4 ‘‘05LI,,4.4,o 57.4 a 9204

c-::— ~‘‘°~‘~‘~%5•’-25. .9909068490 0950 7790

L04,t490ao JO” 0 4000 III?9? Ff9eiOnOcy 9 9100 114

16441650171017281740

09591005102510431055

12291235125513131325

15041510153015481600

095910051025RQ

1055

09397009

343 0955422 7040

I I

i I

.. .5

II, •

122912351255RO

13257330 450 7920 749 ‘3300439 9548 7716 445 9439744? 9558 7734 603 044?

773572057343411

•4

.5

Ii’

NOTES ON BRITISH RAILCONNECTING SERVICES

— untIl 12 Mayfrom IS May

9 — Saturdays II May 1024 September. retlmed loPwIIheIi dsp. 9438.Mlnflordd an’, 0911

FO — Friday. onlySO Saturdays onlyBritith Rail s.rvIcea may besubiect to alierallon at BankHoliday period..

NOTES ON FFESTIN100RAILWAY SERVICES

010 7330 7925 330 625726 ‘43? 7729 439 9725

1440 77 44’ 97

SI. Flestinlog d. 1105 [11 1245 1340 fli~ii 615 1746Tanygrlsiau d. 1111 .{ifl1251 1346 . 1526 1621 1752Tan-y-Bwlch d. 1133 1*11315 1405 ‘.1550 1640 1814Penrhyn d. 1149 4~fl1335 1424 I 1606 1659 1830Mlnffordd d. 1154 .M’1134Q 1429 1611 1706 1835

‘‘a- 747.1000,46 C lies 9530 7790 79559a,m~m 4- 7149 9616 7750 20339616100404 6. 9412 9608 7434 2112

I aOCf 79.6441.06 a - 7450 7533 7909 2130

Porthmadog a., 12091M.L11354 1444j~ii 1719 18481207 1442

0333 450 630 0930 0700 0630 0630 SIlo 7450 “‘0 ‘450 denotes DIESEL HAULED7343 7507 647 5540 ‘Ifl 056? 0640 I 72? 7507 9129 050?0445 609 974? 0940 7214 1409 0940 1324 909 524 g~g flftj TRAINS — ‘ECONOMY’

FARES AVAILABLE.1105 1340 1615 1105 1340 . I • .waConald.rabla reductIon.

U~I for passengera comman.1111 1346 1621 RO RO - • • S cing a FULL RETURNJOURNEY by such trains.1133 1405 1640 1133 1405 .‘ I I 1 The raturn lournay may

1149 1424 1659 1149 1424 • . bemadebyanynln.All other cervIces ar.1154 1429 1704 1154 1429 1’ • 1’ normaIIysteamhauIe~

1205 Isx ?7?9 305 7530 770 07 RD Stop. on request only.7146 7676 7251 249 7679 757 ISiS 975? Paas.ngara wInIng to7325 608 7636 452 609 ‘934 7412 7634 alIght should Inform the353 633 790? 9450 933 9900 7460 4907 guard balora boarding.

Paaaangare wining to17171201 1442 loin should giv, a tint

P6 en, eq

I ±482 ,. ~ I 4’,..2’4~’ . ., f ‘~,,A4O85

-7

I-c.~r’eq

BLAENAUSnow,Icsrs a Na?onaI P.0k FFESTINIOG

17<79.06S’a,eCa.,,

TAN-Y-BWLCH

Maentwroq

handaignal to the drIver.All trains other than the0640 et. Porthmadog wiltalso call On request ciBoston Lodge. PIaa andDduallt.

Every efforl possible willbe made to ansurarunning as ilmetabla butthe Flestiniog Railway wiltnot guarantee advertisedconnections nor theadvertised traction in theeveni ol breakdown orother obetructlon 00services.

Special parties and po*ele charterby arrangement.

Please apply to:FFESTINIOG RAILWAYHARBOUR STATIONPORTHUA000GWYNEDD

TelephonePORTHI6ADOG(0755) 2340/2344

FATHER CHRISTMASEXCURSIONS

A special seovice wIlt operate on21 and 22 December. Deteileavailable from I October.Father Chrisinvas wIll meet thetraini and dIstrIbute presents tothe children,All seats receivable — AdvanceBooking Essential.

R6c~66o,dJ 89/07.07 96,.’ 00,55 ~. R4’i.,~5r

festinioci442Q

Trawstynydd‘0”84’ 0097’ 41.9140

4411? 644 4544707’4±49 .1,61014 001teIla.

by Narrow Gauge Train through the Mountains of Snowdonia

CARBON DATING

Carbon dating is a technique for discoveringthe age of an ancient object, (such as a boneor a piece of furniture) by measuring theamount of Carbon 14 that it contains.

While plants and animals are alive, theirCarbon 14 content remains constant, but whenthey die it decreases to radioactive decay.

The amount, a, of Carbon 14 in an objectI thousand years after it dies is given bythe formula:

a = 15.3 x 0.8861

(The quantity “a” measures the rate of Carbon 14 atom disintegrations and thisis measured in “counts per minute per gram of carbon (cpm)”)

1 Imagine that you have two samples of wood. One was taken from a fresh treeand the other was taken from a charcoal sample found at Stonehenge and is4000 years old.

How much Carbon 14 does each sample contain? (Answer in cpm’s)

How long does it take for the amount of Carbon 14 in each sample to behalved?

These two answers should be the same, (Why?) and this is called the half-lifeof Carbon 14.

2 Charcoal from the famous Lascaux Cave in France gave a count of 2.34 cpm.Estimate the date of formation of the charcoal and give a date to thepaintings found in the cave.

‘—‘if3 Bones A and B are x andy thousand years old respectively. Bone A contains

three times as much Carbon 14 as boneR.

What can you say about x andy?

©Shell Centre for Mathematical Education, University of Nottingham, 1985.Co , I ~7fl’

CARBON DATING... SOME HINTS

Using a calculator, draw a table of values and plot a graph to show how theamount of Carbon 14 in an object varies with time.

0.0

0V

-oCC

CC-oI..CtUC

CzCS

1<

r(1000’sofyears) 0 1 2 3 4 5 6 7 8 9 10 ... 17

a (c.p.m)

Use your graph to read off answers to the questions.

©SheII Centre for Mathematical Education, University of Nottingham, 1985.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Age of object (in 1000’s of years) I

59 (171)

DESIGNING A CAN

A cylindrical can, able to contain haifa litre of drink, is to be manufactured fromaluminium. The volume of the can must therefore be 500 cm3.

* Find the radius and height of the can which will use the least aluminium, and

therefore be the cheapest to manufacture. (i.e., find out how to minimise thesurface area of the can).

State clearly any assumptions you make.

* What shape is your can? Do you know of any cans that are made with this

shape? Can you think of any practical reasons why more cans are not thisshape?

©Shell Centre for Mathematical Education, University of Nottingham, 1985.

60(174~

DESIGNING A CAN... SOME HINTS

* You are told that the volume of the can must be 500 cm3.

If you made the can very tall, would it have to be narrow or wide? Why?

If you made the can very wide, would it have to be tall or short? Why?

Sketch a rough graph to describe how the height and radius of the can have tobe related to each other.

* Let the radius of the can be r cm, and the height be h cm.

Write down algebraic expressions which give

— the volume of the can

— the total surface area of the can, in terms of rand h.(remember to include the two ends!).

* Using the fact that the volume of the can must be 500 cm3, you could

either: — try to find some possible pairs of values for rand h(do this systematically if you can).

— for each of your pairs, find out the corresponding surface area.or: — try to write one single expression for the surface area in terms of r,

by eliminating h from your equations.

* Now plot a graph to show how the surface area varies as r is increased, and

use your graph to find the value of r that minimises this surface area.

* Use your value of r to find the corresponding value of h. What do you notice

about your answers? What shape is the can?

©Shell Centre for Mathematical Education, University of Nottingham, 1985.

61 (17S1

MANUFACTURING A COMPUTER

Imagine that you are running a small business which assembles and sells t~wkinds of computer: Model A and Model B (the cheaper version). You are onl~able to manufacture up to 360 computers, of either type, in any given week.

The following table shows all the relevant data concerning the employees atyour company:

At the moment, you are manufacturing and selling 100 of Model A and 200of Model B each week.

What profit are you making at the moment?* How many of each computer should you make in order to improve this

worrying situation?

* Would it help if you were to make some employees redundant?

©Shell Centre for Mathematical Education, University of Nottingham. 1985.

Job Title Number of people Job description Pay Hoursdoing this job worked

Assembler 100 This job involves £100 36 hoursputting the computers per week per weektogether

Inspector 4 This job involves £120 35 hourstesting and per week per weekcorrecting anyfaults in thecomputers beforethey are sold

The next table shows all the relevant data concerning the manufacture of thecomputers.

Model A - Model B

Total assembly time in man-hours 12 6for each computer

Total inspection and correction time 10 30in man-minutes for each computer

Component costs for each computer £80 £64

Selling price for each computer £120 £88

c.~) (172\

MANUFACTURING A COMPUTER... SOME HINTS

Suppose you manufacture 100 Model A’s and 200 Model B’s in one week:

How much do you pay in wages?* How much do you pay for components?

* What is your weekly income?

* What profit do you make?

2 Now suppose that you manufacture x Model A and y Model B computerseach week.

* Write down 3 inequalities involvingx andy. These will include:

— considering the time it takes to assemble the computers, and the totaltime that the assemblers have available.

— considering the time it takes to inspect and correct faults in thecomputers, and the total time the inspectors have available.

Draw a graph and show the region satisfied by all 3 inequalities:

400

300Number ofModeIBcomputersmanufactured 200(y)

100~ .

0 100 200 300 •400Number of Model A computers manufactured (x)

3 Work out an expression which tells you the profit made on x Model A andy Model B computers.

4 Which points on your graph maximise your profit?

©Sheil Centre for Mathematical Education, University of Nottingham, 1985.~2 (1~7fl\