Which Is Bigger Feel the Fear or the Giant - Lab Report Example

In disciplines such engineering, economics, computer science, social sciences, and naturalsciences, the mathematical modelling application is inevitable. This is because such models offer the most crucial form of knowledge to assist in understanding the system. Some forms of mathematical models include statistical and dynamical models, and differential equations among others. From the problems presented, it is evident that mathematical models begin with real world situations, mostly under the control of some conditions, and at mostly not fully understood environments. The fact that these environments are less understood drivers the need to use observations, understand them and predict the future. Such efforts mostly involve intuition, but often involving examining existing similarities between them and other known and well understood situations. Introduction Mathematical modelling defines the system using mathematical language and concepts to describe its behaviour (Berry and Houston 2-3). The major application is in engineering and natural sciences disciplines, physicists, and computer science among others. Consequently, the use of mathematical models represent crucial elements of the presented system, thus presenting the usable form of knowledge for that system. For instance, modelling “the greenhouse effect” offers the best way to predict future conditions or situations on earth such as extent of carbon monoxide or temperatures. In this report, mathematical models are used to consume real-world data to formulate models that are analysed real world situations to offer conclusions, which after interpretation offer predictions or explanation for any system (Berry and Houston 3). Furthermore, mathematical models offer a means of solving math problems, and representing the association between any two or more variables and that are fundamental to a given problem or situation. Analysis Using differentiation finds the difference between maximum and minimum heights for both the roller coasters. 1. The brochure for the Feel the Fear coaster says that the height of the coaster can be determined by this polynomial model for 12 seconds after the coaster comes out of a loop a) Find the maximum and minimum heights. At stationary points When t = 8 the derivative is negative; hence a maximum point h = = 36 The co-ordinate is (8, 36) When t = 10/3, the derivative is positive hence a minimum point h= = -14.81 The co-ordinate is (3.33, -14.81) Hence minimum height is -14.81 and maximum is 36 b) Draw the graph (by hand) When t = 0; h= Co-ordinate is (0,100) When h = 0; t = 2, 5 or 10 Co-ordinates are: (2, 0), (5, 0) and (10, 0) c) When is the coaster at ground level? Confirm using factor theorem. The coaster is at zero level when h = 0 2 is a factor hence (t-2) = 0 Therefore the quadratic is The coaster is at the ground at 2. The Giant coaster is modeled by The height of the coaster can be determined for the first 12 seconds of the ride by this polynomial (Berry and Houston 10). Calculate maximum and minimum heights using differentiation then draw the graph (by hand) The stationary points occur when When x = 8.2145 the derivative is positive, hence a minimum point When x = 1.7854 the derivative is negative hence a maximum point Minimum height is h = Maximum height is h = When t = 0 The factors are: t = 0, 4, and 11 When t = 0 The co-ordinate is (0, 0) When t = 2 The co-ordinate is (0, 0) When t = 11 The co-ordinate is (0, 0) Where does the ride start? It start at t = 0 Therefore: So the coaster start at zero mark. 3. The ride has 100 metres of fencing to make a rectangular enclosure as shown. It will use existing walls for two sides of the enclosure, and leave an opening of 2 metres for a gate. a) Show that the area of the enclosure is given by: A = 102x – x2 for maximum area : Let length be y Therefore Substitute y; Therefore, the area is: ; b) Find the value of x that will give the maximum possible area. ; At maximum area c) Calculate the maximum possible area Maximum area = 4. Snacks will be provided in a box with a lid (made by removing squares from each corner of a rectangular piece of card and then folding up the sides) You have a piece of cardboard that is 40cm by 40 cm – what dimensions would give the maximum volume? The height is xcm Length is 40-2x Width is (40-2x)/2 since the bottom and up must be identical Therefore: At maximum volume: When x = 20 the second derivative is negative therefore a maximum volume When x = 20/3 the second derivative is negative therefore a maximum volume But x = 20 is not a practical answer . That leaves only 20/3. The dimensions are The maximum volume is: Conclusion This paper evaluates the concept of mathematical modelling in terms of differential equations. The concept is applicable because the variables analysed in the report are continuous, thus varying subject to other continuous variables. In this case, differential equations offer the rate of change of dependent variables subject to independent ones. Since the system is not limited to having only one independent variable, the mathematical model attained from the hypothesis is that which offers a model in terms of leading or advanced order ordinary differential equations of the system. Furthermore, mathematical models normally define a system that needs to maximize a given outcome through provision of some defined inputs. Works Cited Berry, John and Ken Houston. Mathematical modelling. London: Edward Anold, 1995. Read More