This assignment solution addresses a construction mathematics problem, encompassing wave and vector functions, and calculus applications. It begins with an analysis of vibrations in a support beam, calculating amplitudes, frequencies, and time periods for two machines. The solution then delves into vector analysis, determining distances, angles, and vector equations. Furthermore, the assignment explores cost functions, analyzing cost variations, identifying minimum cost points using derivatives, and graphing the functions. Finally, the solution covers calculus applications, including calculating cross-sectional areas and volumes of revolution for a water-retaining structure and a solid hemisphere. The document provides detailed calculations and explanations for each scenario, demonstrating the application of mathematical principles to construction-related problems.