+ + Unit 3 Proportion. Direct proportion. y x means that if x doubles, then y doubles. y enlargement factor original y.

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1 Unit Prprtin Direct prprtin means is prprtinal t (anther sign like =, <, > etc). y x means that if x dubles, then y dubles. We culd write a frmula y kx. This defines a straight line thrugh the rigin and thrugh sme riginal pint xy. All ther pairs f (x,y) pints lie n this line. 6 y + + x, y x It is mre useful t think f an enlargement factr. If x dubles, the enlargement factr =. This makes y duble. S we can write: y enlargement factr riginal y therefre: x y y x The enlargement factr may be >1 r <1. It can even be negative! Example. y = 0 when x = and y x. (a) What is y when x = 0?, (b) what is the frmula y = kx? (a) 0 y

2 (b) T get a frmula, just put a general value x instead f the 0: x 0 y 0 x 5x Other kinds f prprtin. Prprtin ccurs because f a physical frmula F ma hence F a 1 KE mv hence KE v Gm1m 1 Gravitatinal frce F hence F r r We can d any kind f prprtin questin by simply raising the enlargement factr t sme pwer. Example 10 acre fields have a length that is inversely prprtinal t their width, as L w 1. 1 L, which we can write w A field f length 0 yards will have width 100 yards. Hw lng wuld a field be, if the width was 55 yards? 55 L yards Calculatr! Enlargement factr t pwer -1 If yu want a frmula, L w yards yards yards w w Gemetric similarity Sme questins expect yu t identify a suitable prprtin frmula. These usually invlve gemetric similarity. Tw shapes (tw- r three dimensinal, triangles, rectangles, cubids, prisms, pyramids etc) are gemetrically similar if ne is an enlargement f the ther.

3 They lk the same (apart frm size) The rati f any pair f sides in ne shape = the equivalent rati in the ther shape The enlargement factr (side f shape A) t (side f shape B) is cnstant, whichever equivalent pair f sides we chse Each angle in shape A is the same as the equivalent angle in shape B. e.g. a phtgraph is enlarged. h 7 mm Height dubles h 5 mm w 1 mm Aspect rati h 7 9 Width dubles w mm w 1 7 h Aspect rati 5 9 w 7 Aspect rati stays the same Whenever we have an enlargement (i.e. gemetric similarity ) we can say: area length ( and length area ) vlume length ( and length vlume ) vlume area ( and area vlume ) The prprtin frmula must use a crrespnding pair f lengths (e.g. 1, ) and a crrespnding pair f areas r vlumes. It wrks fr any kind f area (area f a shape, crss-sectinal area, surface area). Example A bat f length 5 m has sails f area 1 m. Area length. A gemetrically similar bat is 8 m lng. What sail area des it have? A 8 1 m 0.7 m 5

4 Backwards prprtin If we knw A L ( area length ) what is the prprtin relatin fr length in terms f area? I can square rt each side: A L I can swap sides: L A e.g. An egg with a surface area f 0 cm is 0 mm lng. Hw lng is a gemetrically similar egg with surface area 0 cm? Knwing L A fr any pair f gemetrically similar bjects, I can write: L 0 0 mm. Putting this in my calculatr gives L = 6. mm. 0 Finding a cnstant k Questins ften tell us that a prprtin relatinship beys a frmula such as y and ask us t find the cnstant f prprtinality k. Yu can d these in tw ways: kx r L k A Just substitute a pair f values and slve an equatin fr k, r Write a prprtin frmula using a letter in place f the new x value. Examples: (a) y kx and y = 0 when x =. Find k. 0 Substitute y = 0, x = ; slve 0 k t get k 5 x Since y x I culd write: y 0. This simplifies t y 5x, which is y kx with k 5. (b) L k A and L = 0 mm when A = 0 cm. Find k. It is very sensible t use cmmn units (cm, cm ) t avid cnfusin later (thugh this is nt always k pssible, cnsider F fr gravitatinal attractin) r

5 Substitute L = cm, A = 0 cm ; slve k 0 t get Alternatively write k A L cm 0 0 A k , hence L k A with On the whle, unless the questin asks yu t find k it is much nicer t use the enlargement factr methd. The numbers are exact (did yu rund k? Oh n!) and yu autmatically get the crrect units. Size f a brick questins. This is a kind f prprtin questin where we must chse length, width and height in sme prprtin. Yu can d it in tw ways: a) Writing an equatin and slving it b) Finding the vlume f a basic brick, then enlarging it e.g. The famus architect Arne Jacbsen specified when he designed St Catherine s cllege that the bricks shuld be made lnger and thinner than usual. He wanted a length: height:width in the rati 11::. If each brick has a vlume f 1000 cm, what are the dimensins? (a) x x 11x I will define each length as a multiple f an unknwn length x. The 11x, x, x multiples make them in the crrect prprtin. Nw we write an equatin: 11x x x 1000 cm 1x 1000 cm x cm 1 The sides are then 1.6 cm, 5.9 cm, 7.9 cm. (b) Alternatively I can imagine a basic brick with sides 11 cm, cm, cm, find its vlume, then scale it up cm 5

6 Knwing length vlume, I need t enlarge each side by factr f The sides are then cm 1, cm, cm