Finding Minima and Maxima Using Mathematical Modeling - Coursework Example

Math report Finding minima and maxima using Mathematical modeling College In this report mathematical modeling methods are going to be used to find the maximum and minimum heights for two roller coasters namely Feel the Fear and the Giant roller coaster. The same methods are to be applied in finding the maximum possible area of a fence enclosure and the dimensions of a snack bot those that yield maximum volume. A step by step process is going to be used in the presentation of this report and the equations as well as the problems are going to be elaborately explained. Introduction Mathematical modeling is the process of describing a system using mathematical concepts and mathematical language. Mathematical models are used in many areas especially in the natural sciences where they are used to model the behavior of different agents, to explain and to predict their behavior. The methods are used in economics, physics, and chemistry, geology, in statistics, in operational research and in other disciplines. In this report, the method of differentiation is going to be employed extensively in finding most of the solutions. Also, the graphs of the functions where needed will be drawn. Q1 Analysis Feel the fear The maximum and minimum heights for feel the fear can be determined by the given Polynomial and for the first 12s To find the maximum and minimum points for this polynomial we have to solve for the first and second derivatives. These are to be gotten as follows; Now we need to solve for as this gives us the maximum and minimum points By factorization The two possible values of t Replacing in the polynomial we get that Meters Meters Now which of these is the maximum? For t=8 we have that For t =3.3 we have that Hence the maximum height for the roller coaster is 36 meters while the minimum is – 14.815 meters. The difference between the two is 50. 815 meters Graphing the polynomial To graph the polynomial, we need to find points that are covered by the curve. Most appropriate for our case are t=0 t=12 and the maximum and minimum points. For t=0 we have that Meters for t=12wehave that Meters 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, meters When t = 12, meters When h = 0, For h =0 we have that The resulting graph The Giant rollercoaster According to the information given, the height of The Giant can be determined through the same process Calculating the maximum and minimum heights 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 To solve for the turning point, we have to equate h = 0 ie Solving by completing the square method Solving the equation for turning points The two values of t Replacing values in the second derivative to find the maximum and minimum heights we get For t = 1.785 For t= 8.215 Accordingly, t = 1.785 gives maximum height of And t = 8.215 gives the minimum height of The difference between the maximum and minimum heights is therefore given as -96.434 + 36. 434 and which is 132.868. Graphing the polynomial To graph the polynomial, we need to determine several things. These are the points when the height is 0, t =12, t = 0 and the maximum and minimum points. When t = 0, When t = 12, When h = 0, 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 above yields the following graph Area of the enclosure 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 as shown below we are to show that the area of the enclosure is given by: A = 102x – x2 further we are to calculate the maximum possible area. 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 We first must get the first and second derivative as shown Solving the equation for turning points Therefore x at 51 meters gives the maximum area and which is given as Square meters Finding the maximum volume of the snack box 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. From the above, 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 . Accordingly, the volume of the snack box is given by Now to find maximum volume we must first get the derivatives of V and which are give as follows Now solving the equation to get the turning points Putting the value of x in for the two instances of x 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 To arrive at the many conclusions, the particular mathematical modeling techniques used was differentiation. It mostly involved finding the maxima and minima of the various mathematical expressions that were already given or else arrived at. The results show that the maximum and minimum heights for feel the fear roller coaster are 36 meters and -14. 1815 meters respectively. The difference between the two was found to amount to 50.815 meters. On the other hand, The Giant roller coaster is 36. 434 at its maximum and -96.434 at its minimum this roller coaster starts at the ground level. For the case of the enclosure the maximum possible area was found to be the2601 square meters where the width is 51. Finally the dimensions of the snack box that yield maximum volume are as follows 6.67 cm x 13.33 cm x 26.67 cm. in this case the maximum volume is 2370.37 cubic cm. References Courant, R. and McShane, E. (1937). Differential and integral calculus. New York: Interscience. Finney, R., Thomas, G. and Weir, M. (1994). Calculus. Reading, Mass.: Addison-Wesley Pub. Co. Read More