Which Is Bigger Feel the Fear or The Giant - Coursework Example

Math Report 2 – March Which is bigger Feel the Fear or The Giant? Your College The mathematical modelling is used to find the difference between maximum and minimum height for the Feel the Fear and The Giant roller coasters. It is also used for finding maximum possible area of fixed fence enclosure and the dimensions of a snack box that would give the maximum volume. The findings indicated that the maximum and minimum height of the Feel the Fear roller coaster are 36 meters and -14.815 metres. The difference between maximum and minimum heights for the Feel the Fear roller coaster is 50.815 metres. The findings also indicated that the maximum and minimum height of The Giant roller coaster is 36.434 meters and -96.434 metres. The difference between maximum and minimum heights for The Giant roller coaster is 132.868 metres. The maximum possible area of the rectangular enclosure found to be 2601 square metres when the width is 51 metres. The dimensions of the snack box that would give the maximum volume found to be 6.67 cm x 13.33 cm x 26.67 cm. The maximum volume of the snack box found to be 2370.37 cubic cm. Math Report 2 – March 2015 Which is bigger Feel the Fear or The Giant? Introduction Mathematical modelling is a method for solving problems mathematically. A mathematical model is a mathematical representation of the relationship between two or more variables relevant to a given situation or problem. Mathematical modelling is used to investigate important questions about the observed world, to describe real-world incident, to explain real-world incident, to test ideas and to make predictions about the real-world. In this report, mathematical modelling will be used to answer the real-world questions. The formulation, analysis (using differentiation, maxima and minima), interpretation and test will be done to answer the real-world questions. The mathematical modelling will be used to find the difference between maximum and minimum height for the Feel the Fear and The Giant roller coasters. It will be also used for finding maximum possible area of fixed fence enclosure and the dimensions of a snack box that would give the maximum volume. Analysis Height difference of Feel the Fear According to the brochure for the Feel the Fear, the height of the roller coaster can be determined by below given polynomial model for 12 seconds after the coaster comes out of a loop. Maximum and minimum heights For finding maximum and minimum heights of the Feel the Fear coaster, firstly, we need to find the first and second derivatives. The first and second derivatives of the polynomial model are Solving the equation for turning points. Putting the value of t in to for finding which value of t gives maximum height and which value of t gives minimum height. When t = 8, When t = 10/3, Therefore, t = 8 seconds gives maximum height and t = 10/3 gives minimum height for the Feel the Fear coaster. metres metres The maximum height of the Feel the Fear roller coaster is 36 meters and the minimum height is about -14.815 metres (14.815 below ground level). Therefore, the difference between maximum and minimum heights for the Feel the Fear roller coaster is about 50.815 metres. Graph of the polynomial model For graphing the polynomial model, we need to find the coordinates when the height is 0 and the heights of the roller coaster when t = 0 and t = 12 seconds. When t = 0, metres When t = 12, metres When h = 0, The graph of the polynomial model is given below showing maximum and minimum heights for the Feel the Fear roller coaster. The time when the coaster is at the ground level The time when the coaster is at the ground level is at t = 2 seconds, 5 seconds and 10 seconds. Putting values of t = 2, 5 and 10 in . The remainder is zero for t = 2, 5 and 10 seconds. Therefore, according to the Factor Theorem, it is confirmed that the factors (t – 2), (t – 5) and (t – 10) are factors of . Height difference of The Giant The height of The Giant coaster can be determined by below polynomial model for the first 12 seconds of the ride. Maximum and minimum heights The first and second derivatives of the polynomial model are Solving the equation for turning points. Solving by completing the square. Putting the value of t in to for finding which value of t gives maximum height and which value of t gives minimum height. When t = 1.785, When t = 8.215, Therefore, t = 1.785 seconds gives maximum height and t = 8.215 gives minimum height for The Giant coaster. metres metres The maximum height of The Giant roller coaster is about 36.434 meters and the minimum height is about -96.434 metres (96.434 below ground level). Therefore, the difference between maximum and minimum heights for The Giant roller coaster is about 132.868 metres. Graph of the polynomial model For graphing the polynomial model, we need to find the coordinates when the height is 0 and the heights of the roller coaster when t = 0 and t = 12 seconds. When t = 0, When t = 12, When h = 0, The graph of the polynomial model is given below showing maximum and minimum heights for the Feel the Fear roller coaster. Where does the ride start? The ride start (t = 0) at the ground level. Maximum area of the enclosure The ride has 100 metres of fencing to make a rectangular enclosure as shown and it uses existing walls for two sides of the enclosure, and leave an opening of 2 metres for a gate. Area of the enclosure The width, W of the enclosure is x m. Therefore, the length, L of the enclosure will be m The area of the enclosure is given by: Value of x that will give the maximum possible area The first and second derivatives of the A are Solving the equation for turning points. Therefore, the value of x that will give the maximum possible area is 51 metres. Maximum possible area square metres The maximum possible area is 2601 square metres. Dimensions of the box that will give a maximum volume The 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 as shown in below figure. The box is made with a piece of cardboard that is 40 cm by 40 cm. The height, H of the snack box is x cm. The length, L of the snack box is 40 – 2x. The width, W of the snack box is . Therefore, the volume of the snack box is given by: The first and second derivatives of the V are Solving the equation for turning points. Putting the value of x in When x = 20, When x = 20/3, The maximum volume is when x = 20/3 cm. Therefore, the dimensions of the snacks box are Height, H = x =20/3 cm = 6.67 cm Length, L = 40 – 2x = 40 – 2(20/3) = 80/3 cm = 26.67 cm Width, W = 20 – x = 20 – 20/3 = 40/3 cm = 13.33cm The dimensions that would give the maximum volume are 6.67 cm x 13.33 cm x 26.67 cm. The maximum volume is cubic cm Conclusion I used formulation, differentiation, maxima and minima to find the answers. The maximum and minimum height of the Feel the Fear roller coaster are 36 meters and -14.815 metres. The difference between maximum and minimum heights for the Feel the Fear roller coaster is 50.815 metres. The Feel the Fear roller coaster is at ground level at 2 seconds, 5 seconds and 10 seconds. The maximum and minimum height of The Giant roller coaster is 36.434 meters and -96.434 metres. The difference between maximum and minimum heights for The Giant roller coaster is 132.868 metres. The Giant roller coaster ride starts at the ground level. The maximum possible area of the rectangular enclosure is 2601 square metres when the width is 51 metres. The dimensions of the snack box that would give the maximum volume are 6.67 cm x 13.33 cm x 26.67 cm. The maximum volume of the snack box is 2370.37 cubic cm. Reference Berry, John; Houston, Ken (1995-06-17). Mathematical Modelling (Supporting Early Learning) (p. 1). Elsevier Science. Kindle Edition. Read More