The first question, Q1, requires determining which roller coaster, the Giant or Feeling the Fear modelled by and respectively, is bigger. In order to achieve this, the minimum and maximum heights of bother roller coasters is determined, a graph is then drawn, and finally, for the Feeling the Fear coaster, a determination is made of when the coaster is at ground level, while for the Giant coaster, a determination is made of where the ride begins. For the Feel the Fear coaster, a graph is made and the height of the coaster at time, through to is determined.According to the graph, the Feel and Fear roller coaster is at ground level when this happens at three different instances, when. Given the equation and using the factor theorem in which we divide by, we can check whether or not the coaster reaches the ground level at instanceThe first part of the question involves illustrating that the area of the enclosure can be calculated by. In this case, we assume the width to x meters, and the length to meters. In this regard, the area of the enclosure is given by:The piece of cardboard is 40cm by 40cm; the goal is to determine what dimensions would give a box of maximum volume if squares from each corner of the cardboard are removed and then the sides are folded up. To do this, we assume that the length of the cardboard after removing squares of side x will be:Using mathematical modelling, we have been able to solve four different real world problems, albeit in theory. Using the differentiation and quadratic formula, we have been able to determine the maximum and minimum heights of each roller coaster; maximum height of 36 at t=8, and minimum height of 14.8 at, for the Feel and Fear coaster. Maximum height of 36.4 at t=1.8, and minimum height of -96.4 at, for the Giant coaster; the ride begins at h=0. For the Feel and Fear Coaster, the report indicates that the coaster is at ground level at times, t=2, t=5, and t=10; this successfully confirmed using the factor theorem. For the third question, the analysis confirms, using mathematical modelling, that the area of the enclosure is given by A = 102x – x2. Through differentiation, we have successfully determined the value of x that would give the maximum value to be and thus, the maximum possible area to be. Finally, through mathematical modelling, we determine, for Q4, that for a cardboard of 40cm by 40cm, the dimensions that would give the maximum volume are height,
Barnes, B., & Fulford, G. . R. (2008). Mathematical Modelling with Case Studies: A Differential Equations Approach. Boca Raton, FL: CRC Press.
Berry, J., & Houston, K. (1995). Mathematical Modelling (Supporting Early Learning) (Kindle Edi., p. 24). New York, NY: Elsevier Science.
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