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 develop model using related mathematical techniques to successfully ensure client requests about pool size and create an aesthetic appeal adhering to a 1:2 ratio between pool surface area and backyard area. In addition, 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 no more than three of polynomials. This task included formulation of the three polynomials: linear f(x)=mx=c, quadratic f(x)=ax^2+bx=c, cubic f(x)=ax^3+bx^2+cx+d, one exponential:f(x)=e^x and two trigonometric: 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, polynomial, and point of intersection were used to set the upper and lower limits of the integrals, area under the curve was done by the mathematical equation on 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.
Documentation of relevant observations
The report displays the practical application of mathematics in solving real-world problems, particularly in economic profit analysis and designing when homeowner sell their house.
1. The backyard the 5% margin of error is 9.5m^2 smaller or larger.
2. Fulfilling 6 different functions including linear, quadratic, cubic, exponential, sine, and cosine, that construct the pool design adheres to the desired dimensions and aesthetics.
3. In the cartesian plane in Desmos, the x and y represent m^2distance
4. Using definite integral the three integrative functions and the surface area of the pool was calculated.
5. Trigonometric function forms a unique shape in pool but not inherently possess their own curves and the exponential function forms the best curved surfaces.
6. The allocated shape and backyard is rectangle the surface area is 〖190m〗^2
Solve
Three integrative functions (manual calculations)
∫_POI^POI▒〖top function-bottom function〗×dx
Cubic-Expotential (radain)
In the manual calculation, three integrative functions were calculated employing the formula ∫_POI^POI▒〖top function-bottom function〗×dx to find the area between two curves. Polynomial (cubic) and exponential both equations were integrated and defined within the domain (0.97≤x≤1.24), where functions intersect. By substituting the given domain values into the formula, and computing the integral area under each curve. For the cubic polynomial, the area was calculated separately from the exponential function. The final area between the curves was determined by subtracting the area of the cubic polynomial from the area under the exponential, resulting in a value of 0.63m^2 rounded to 2 decimal points.
1/3 x^3+3x^2-x+2.5[0.97≤x≤1.24]-7e^(2.5-3x) [0.97≤x≤1.24]
∫_0.97^1.24▒〖1/3 x^3+3x^2-x+2.5dx〗
=(1/3 x^4)/4+(3x^3)/3-x^2/2+2.5x]_0.97^1.24
=(1/12 (1.24)^4+(3(1.24)^3)/3-(1.24)^2/2+2.5(1.24))-(1/12 (0.97)^4+(3(0.97)^3)/3-(0.97)^2/2+2.5(0.97))
∴1.493844412m^2
∫_0.97^1.24▒〖7e^(2.5-3x) 〗 dx
(7e^(2.5-3x))/(-3)+C]_0.97^1.24
=((7e^(2.5-3(1.24) ))/(-3)+C)-((7e^(2.5-3(0.97) ))/(-3)+C)
∴0.8596468605m^2
Area between curves: 1.493844412-0.8596468605=0.6341975515m^2
sine – cosine-radian
In this calculation, Area 4 was determined to be enclosed between a sine function and a cosine function using the definite integral formula∫_POI^POI▒〖top function-bottom function〗×dx. To satisfy the three integrative functions, two trigonometry functions were subtracted from one another. Firstly, integration was performed for each function individually, and the difference was computed to find the total area between the curves. This was done over the specified domain 3π/2≤x≤13.57. Then the collected areas of the individual fractions, top function and bottom function, were subtracted resulting =62.27m^2 total area between the curves.
(0.5 sin(x)+8)[3π/2≤x≤13.57]-(-1cos(x + π/2)+1)[3π/2≤x≤13.57]
=∫_(3π/2)^13.57▒〖0.5 sin(x)+8 dx〗
=-0.5 cos(x)+8x+c]_(3π/2)^13.57
(-0.5 cos(13.57)+8(13.57))- (-0.5 cos(3π/2)+8(3π/2))
∴70.59226576m^2
=∫_(3π/2)^13.57▒〖-1cos(x + π/2)+1dx〗
=-1 sin(x+ π/2)+1x+c]_(3π/2)^13.57
(-1 sin((13.57)+ π/2)+1(13.57))-(-1 sin((3π/2)+ π/2)+1(3π/2))
8.320366288m^2
Area between curves: 70.59226576-8.320366288=62.27189947m^2
Desmos calculation
The design of the pool was deliberately researched and planned beforehand after total area and shape constraints were predefined by the teacher. Extensive research indicated a strong demand among Brisbane’s great population for a pool featuring a curved top and bottom section with sharp, edgy sides. The particular selected design, a unique and striking flag-shaped pool, was deemed suitable for all family households. The design was achieved by utilising a sine function to create the upper section of the pool, a cosine function for the curved bottom section, and an exponential function for the transition to sharp edges, continuing with cubic and quadratic elements. On the opposite side, a linear function was used to form the desired state. Thus, the final design profiles were finalised through the thorough use of a combination of six different functions, generating in the integration of a flag-shaped pool.
give me some soome reecommdnation can be done for furture
Based on the information provided, here are some recommendations for future improvements:
1. Include more detailed explanations of the mathematical concepts and techniques used in the report. This will help readers with varying levels of mathematical understanding to follow along more easily.
2. Provide clear justifications for why specific mathematical functions were chosen for different parts of the pool design. This will help readers understand the reasoning behind the design choices and how they contribute to the overall aesthetic appeal.
3. Consider including visual representations of the pool design, such as graphs or diagrams, to make it easier for readers to visualize the final product.
4. Conduct further research on market preferences and trends in pool design to ensure that the chosen design is truly unique and appealing to potential buyers. This could involve surveying homeowners or consulting with industry professionals.
5. Explore the use of other mathematical techniques or models that could enhance the pool design, such as optimization algorithms or simulation software. This could help refine the design and ensure that it meets all client requirements, including the 1:2 ratio and 5% margin of error.
6. Consider conducting a case study or real-world application of the pool design to validate its effectiveness and appeal. This could involve working with a client to implement the design and gathering feedback on its performance.
Overall, continuously updating and refining the pool design process using feedback and new mathematical techniques will ensure that the final product meets client expectations and remains competitive in the market.