Maths Coursework Trays Essays

Maths Coursework Trays Essays Maths Coursework Trays Essay Maths Coursework Trays Essay In this coursework candidates were given a task entitled Trays. The task consisted of a shopkeepers statement upon the volume of a tray which was to be made from an 1818 piece of card. The shopkeepers statement was that, When the area of the base is the same as the area of the four sides, the volume of the tray will be maximum. By saying this, the shopkeeper basically meant that when the area of the base of the tray is equal to the total area of the sides the volume of the tray will be at its highest. We were told to investigate this claim.Plan.1. I will investigate the different sizes of tray possible from an 1818 piece of card.2. After gaining my results I will then put them in a table.3. I will try to spot any patterns from my table.4. I will express any patterns or other formulae in mathematical notation.To investigate the different volumes given by different trays, I first decide to cut the corners in ascending order from 1-8. (The longest possible corner could only be 8 as after this there would be no base.) After this I worked out the formula needed to work out the volume for the various trays. For the corner size 11 the way I worked out the volume was 16x16x1 which equalled 256cm. Thus the formula to work out the volume for a tray made by an 18x18cm card is (n 2X) x X. In this formula the letter X represents the size of the corner. I tried my formula for the corner length of 2cm,(18- 2 x 2) x 2(n 2 x X) x X(n 2 x X) x XI take off two the corners from each side as the card is square.After finding out the formula I worked out the volume for the remaining trays.CornersVolume (cm)16x16x11x125614x14x22x239212x12x33x343210x10x44x44008x8x55x53206x6x66x62164x4x77x71122x2x88x832From my table I can see that the highest volume for a tray made by 18x18cm card is 432 cm this volume is reached if the corners cut are 3cm x 3cm. I can also see that the volume of the tray rises as the length of each corner rises until the corner size goes over 3. After this the volume starts to decrease as the size of the corner increases.After working out the volume for the trays I went on to work out the area of the bases of the trays along with the areas of the sides of the trays. I worked out the area of the base of the tray by finding the size of the side after the corner had been cut off and then square this number. For example to find out the area of the base of the tray where the corners were 1x1cm ,I first found out the size of the sides which were 16 and squared it. The answer was 256cm . The formula for this was (n 2x) which out would be 18 (n) minus 2 times 1(x) squared. I than proceeded to work out the area of the sides, which would be essential in proving that the shopkeeper is right. To work out the are of the sides of the tray I used the formula 4x (n- 2x). Here again the n represents the size of card 18cm. The x represents the size of the corner. You have to times your answer by four as there are four sides. To work out the area of the sides for a corner sized 1x1cm the calculations would be:4x (n 2x)4 x 1 (18 2 x 1)4 ( 16 )64cmCornersVolume cmArea of base cmArea of sides cm1x1256256642x23921961123x34321441444x44001001605x5320641606x6216361447x7112161128x832464From my results I can see that in regards to the area of the base, the area lowers as the corner size is increased. However the area of the sides increases as the size of the corner increases until the corner reaches the size 44 cm. After this the areas are repeated in reverse order.I then looked at my results to see whether any areas matched.I noticed that for the corner size of 3x3cm the areas matched as the area of the base was 144cm and the area of the sides was 144cm . I also noticed that the highest volume for a tray made from an 18 by 18cm piece of card was 432cm which also derived from the corner size 3cmX 3cm. I can thus make the conclusion that the shopkeeper is right.However to make sure that 432cm was the highest possible volum e available from an 18 by 18 piece of card I decided to use decimals. I decided on investigating corners of 2.9cm and 3.1cm . I used the same formulas.CornersVolumeArea of base cmArea of sides cm2.92.9431.636148.84141.52334321441443.13.1431.64139.24146.32From these set of results I can see that the corner size of 3cm has a higher volume than the corner 2.9cm or the corner 3.1cm. Also the areas of the sides and of the base only match when the corners cut out are equal to 3cm. I can therefore make the conclusion that to get the maximum volume from an 18cm by 18 cm card you need to have to cut out corners of three centimetres.I decided to see whether the shopkeepers theory was correct on different sized square cards. The card of which the trays would now be made will be sized 20 x 20 cm. I transferred the same formulae for the 18 x 18cm card. I recorded the following results:CornersVolume cmArea of base cmArea of sides cm1x1324324722x25122561283x35881961684x45761441925x55001002006x6364 621927x7294361688x8256161289x9162472You can see from the results that they are very similar to those which were recorded on the 18 by 18cm card. However there is one main difference, the maximum volume is not given when both the areas of the base and area of sides is equal. Thus I graphed the area of the sides against the area of the base.You can see from my graph that the two area values crossed between 3 and 4 consequently the highest value lay between these two numbers if the shopkeeper was right.CornerVolumeArea of baseArea of Sides3.05589.2905193.21169.583.1590.364190.44171.123.15591.2235187.69172.623.2591.872184.96174.083.25592.3125182.25175.53.3592.548179.56176.883.35592.5815176.89178.223.4592.416174.24179.523.45592.0545171.61180.783.5591.51691823.55590.7555166.41183.183.6589.824163.84184.323.65588.7085161.29185.423.7587.412158.76186.483.75585.9375156.25187.53.8584.288153.76188.483.85582.4665151.29189.423.9580.476148.84190.323.95578.3195146.41191.1845761441924.1570.884139.241 93.524.15568.0935136.89194.224.2565.152134.56194.884.25562.0625132.25195.54.3558.828129.96196.084.35555.4515127.69196.62I conclude from my results that the shopkeepers statement is not true on a 20x20cm card.