The swimming pools are one of those extra features that can contribute to the "making" factor when the owner is selling the house. The Problem Solving and Modelling Task (PSMT) aims to utilise variety functions such as trigonometry, exponential, and polynomial functions to custom-design a unique pool for a specific backyard that has been specified. The equations were sketched and checked by a graphic calculator and Desmos (Desmos, 2023). The purpose of this report aims to derive the development and evaluation of the model using related mathematical techniques to successfully ensure client requests about pool size and create an aesthetic appeal consisting of a condition of carter for families of all types. Adhering to a 1:2 ratio, the pool’s surface must be within a 5% margin of error relative to the allocated backyard space.
Formulate (Mathematical concepts and techniques)
This report has developed to use at least 6 different functions to create the pool shape, under condition of o more than three of polynomials. This task included formulation of the polynomial: linear f(x)=mx=c, quadratic f(x)=ax^2+bx=c, cubic f(x)=ax^3+bx^2+cx+d, exponential:f(x)=e^x and trigonometry: sin f(x)=AsinB(x-C)+D, cos f(x)=AcosB(x-C)+D.
When finding area between the curves the function ∫_a^b▒〖f(x)-g(x)〗×dx was used, Area top – Area bottom even when finding graphs underneath the x axis of intersecting with the x-axis.
To enhance the connectivity between the six distinct functions, intended technology contributed to the custom-flagged design pool. Desmos was employed to calculate the pool’s area, ensuring a 1:2 ratio within a 5% margin of error. Moreover, manual calculation of three integrative functions, such as trigonometry, exponential, and polynomial, was done by the graphics calculator with Word.
Documentation of appropriate assumptions
It is essential that all calculations assume that the solution meets the appropriate criteria. The 6 functions mixed combination of trigonometry, exponential, and polynomial, no more than 3 polynomials make sure meeting the criteria of make factor in competitive market unique design is required. All the functions that were used to generate the pool must pass the vertical line test because this would than would not match the output (y-value) value for each input (x-value) value and approved as invalid function. The pre-designed pool shape that will create aesthetic appeal is in Brisbane, Australia, taking into account the region’s specific preference. It maintains a 1:2 ratio and within a 5% margin of error, maximising the use of available space in the allocated backyard. This ensures that the total area of the pool complements the backyard without extending beyond the boundaries.
give some documented observaton on this