Mansfield High, Mansfield

MATH 123

Introduction 1.1 Aim The aim is to create a swimming pool with at least three curves, find the volume of water that is going to be used to fill up the pool, and determine the cost of the water as charged by the city of Gold Coast. 1.2 Method Microsoft Word was used to sketch an enclosed curve shape that would later be my prototype pool design. The upper two curves were reshaped to be the same height, so it would be easier to measure the overall height of the model (see appendix A). The enclosed design was rescaled on word to a height of 12cm and a width of 20cm. 5 sections were determined on the pool to be integrated (see appendix B). The graphing application ‘Desmos’ was used to rescale the pool design to the correct scale. Since the scale for this assignment is 1:200, the model’s final height and width in Desmos was 2400cm by 4000cm. The pool was then copied and pasted into a picture file, where the borders were then cropped to the edges of the pool, to reduce error in scaling. The image of the pool was then placed into Desmos and was rescaled to fit the final measurements of the pool (40m width, 24m height). The pool was then repositioned to have half of the model above the x-axis, and the other half below. Digitised points were then plotted around the outline of the pool model in Desmos (with a plot width of 1m). After plotting the points around the five sections of the pool (see appendix B), three general functions per section were chosen that were thought to have the highest accuracy to the plots. From that, 5 best functions for the pool sections were determined: each function having the best R 2 value (Frost, 2022) as well as the best continuity with adjacent functions. The five final functions were then integrated through the fundamental theory of calculus: both by hand, and using Desmos. The areas of each section were then totalled, and further evaluated through trapezoidal rule: area= h 2 [E+2 M] , where h is the strip width (calculated by h= b−a n , where b and a are the upper and lower bound respectively and n is the number of strips), and E & M are the sums of the endpoints and midpoints respectively in the table (BYJUS Learning, 2020). The area of the base of the pool was then multiplied by the height of the pool to obtain the total capacity of the pool and therefore find the total volume and cost of the water to fill up the pool. The pool was then evaluated on its reasonableness and design, and refinements were made to improve the pool model. 2.0 Considerations 2.1 Observations and Assumptions The following observations were made before starting the pool sketch: at least 3 curves must be in the pool; 3 different types of functions must be present in the pool; the scale is to be 1:200; the domain of each section must be stated to have a good fitting function; and the depth of the pool is 1.5m (Taylor, 2022). It was also observed during the pool modelling, that some of the residuals went above or below the x-axis, indicating an over/underestimation of the plot in relation to the predicted regression curve. The following assumptions were also made before starting the pool sketch: the pool base has uniform depth; the pool base and walls are perpendicular, with no curved edges; there are no stairs in the pool model; periodic functions that were used