F23-Tut-03

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 3 1|3 Name: ID: Date: 1. The figure below shows the flow of traffic along one-way roads during rush hours in a study area. The arrows indicate the direction of traffic flow. The average vehicle flow (number of vehicles per hour) are shown outside the study area. The variables 𝑥 1 , 𝑥 2 , 𝑥 3 , and 𝑥 4 represent the flows inside the study area. The flow of traffic is controlled by traffic lights installed at each of the four intersections. Problem from Tan, Soo T., Finite Mathematics for the Managerial, Life, and Social Sciences, 9th Ed. The avenues (5 th and 6 th ) have a maximum capacity of 2000 vehicles per hour. The streets (4 th and 5 th ) have a maximum capacity of 1000 vehicles per hour. Traffic congestion within the study area is caused when the roads exceed their maximum capacities. To avoid congestion, all traffic entering an intersection must also leave that intersection. Applying this condition to each intersection in the study area yields: Intersection A: 1200 + 300 = 𝑥 1 + 𝑥 4 Intersection B: Intersection C: Intersection D: a) Determine the system of equations that governs uncongested traffic flow in the study area. b) Determine the RREF of the augmented matrix representing the system. c) Determine the general solution for 𝑥 1 , 𝑥 2 , 𝑥 3 , and 𝑥 4 , making sure to incorporate the constraints that ensures no traffic congestion will occur. d) Suggest two possible solutions that ensure that no traffic congestion will occur in the study area. Entering Leaving

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 3 2|3 2. The unbalanced chemical equation shown below describes sugar fermenting to produce carbon dioxide and ethanol. 𝐶 6 𝐻 12 𝑂 6 → 𝐶𝑂 2 + 𝐶 2 𝐻 5 𝑂𝐻 One way to balance a chemical equation is by introducing variables and forming a system of linear equations as illustrated below. 𝑋 1 𝐶 6 𝐻 12 𝑂 6 → 𝑋 2 𝐶𝑂 2 + 𝑋 3 𝐶 2 𝐻 5 𝑂𝐻 Element Left Side Right Side Carbon 6𝑋 1 = 𝑋 2 + 2𝑋 3 Hydrogen 12𝑋 1 = 6𝑋 3 Oxygen 6𝑋 1 = 2 𝑋 2 + 𝑋 3 Balance the chemical reaction above by solving the system of equations.