Math 104b Problem 5 Group 5 Eleanor Browne, Nicola Wong, Paul Culkin Information A plain wall, of dimensions 17.4m by 3m, needs to be covered with wallpaper. The wallpaper comes in rolls of 12.1m long and 0.60m wide. We were told that there are two designs of wallpaper. The first has a spaceship pattern on it, which repeats itself vertically every 11cm; the second has a flowery pattern, which repeats itself vertically every 7cm. The method of wallpapering is to start at the top left of the wall and paste a strip of wallpaper down to the bottom of the wall. The second strip starts at the top of the wall again, adjacent to the first strip. A section must be cut off the roll so that the second strip starts at the same point in the pattern as the first one did. Task One We needed to show that 10 rolls of spaceship wallpaper were needed to paper the whole wall. So, to complete this part of the task: We first worked out how many repeats of the pattern we would need for each strip, as this would, in turn tell us the length of wallpaper needed for each strip. The spaceship pattern repeats itself every 11cm so, 300 ÷ 11 = 27.272727 This tells us we need ( 28 × 11)cm for each strip. That is 308cm , where 8cm goes to waste. The rolls of wallpaper are 12.1m long so now we can work out how many strips can b used per roll, 12.1 ÷ 3.08 = 3.92857... This tells us each roll of wallpaper can b used for 3 strips only. The rolls of wallpaper are 0.60m wide and we have just found that each roll can be used for 3 strips, so we can therefore calculate that each roll can be used to cover 1.8m in width. We can now work out the total number of rolls needed to cover the whole wall: 17.4 ÷ 1.8 = 9 2 3 Therefore we need ten rolls of spaceship wallpaper to complete the wall. We also calculated that for each spaceship patterned roll we would waste 3.1m Math 104b Problem 5 Group 5 Eleanor Browne, Nicola Wong, Paul Culkin Task Two To find out how many rolls of the flowery patterned wallpaper we will use the exact same procedure: The flowery pattern repeats itself every 7cm and 300 ÷ 7 = 42.8571... . This tells us 1cm goes to waste. The number of strips that can be used per roll is: 12.1 ÷ 3.01 = 4.01993... that we need ( 43 × 7 )cm . That is, 301cm where The rolls of wallpaper are 0.60m wide and we have just found that each roll can be used for 4 strips, so we can therefore calculate that each roll can be used to cover 2.4m in width. The total number of rolls needed to cover the wall is: 17.4 ÷ 2.4 = 7 1 4 Therefore we need eight rolls of flowery wallpaper to complete the wall. For each roll of flowery wallpaper we worked out that 10cm would go to waste. Task Three To work out a generalised formula for any given length of repetition (d) we have split the procedure we used to work out parts 2 and 3 onto 3 steps as follows… First we want to find how much paper is “used” (including waste) for each 60cm strip. This is found using the following formula….   300      + 1 × d  d   The next step is to find how many full strips we can get out of a 12.1 meter length role. Given by….    300    1210 ÷    + 1 × d   d     Finally we need to divide this value into the number of strips wide the wall is (which remains fixed at 1740/60 = 29) giving….     300     29 ÷ 1210 ÷    + 1 × d      d     Checks To check our formula, we substituted the values of d for which we had already worked out the number of sheets needed (i.e 11 for the spaceship pattern and 7 for the flowers), and verified it did in fact come out with the same answers of 10 and 8 respectively.