Math Problem Statement

Each number on the diagram represents the average number of cars per day entering (blue) or leaving (red) these one-way streets. The arrows indicate the direction of traffic flow. NCDOT aims to model the hourly traffic flows along these roads as part of their plans to update the current traffic control measures (stop signs, traffic lights, etc.) at the intersections. Define x, u, v, and z to be the traffic flow in cars per day between intersections along Jones St., Wilmington St., Edenton St., and Salisbury St.,respectively. Furthermore, define y to be the total cars per day exiting the intersection of Jones St. and Wilmington St. to the east along Jones St. (a) [10 points] Identify the constraints in this problem, and use them to set up a system of linear equations that describes the traffic flow along each street. Your answer should include: i. A list of constraints along with an explanation for each constraint. ii. The linear system of equations that models this scenario. (b) [5 points] Use the Gauss-Jordan reduction method to solve the linear system obtained in part (a). Your answer should include: i. An augmented matrix representation of the system of equations in part (a). ii. The final augmented matrix obtained after applying the Gauss-Jordan reduction method. iii. The solution to the linear system. (c) [5 points] Use Google Maps, a physical map, or your personal experience/intuition to carefully examine the set of intersections under consideration. Then answer the following:

Each number on the diagram represents the average number of cars per day entering (blue) or leaving (red) these one-way streets. The arrows indicate the direction of traffic flow. NCDOT aims to model the hourly traffic flows along these roads as part of their plans to update the current traffic control measures (stop signs, traffic lights, etc.) at the intersections. Define x, u, v, and z to be the traffic flow in cars per day between intersections along Jones St., Wilmington St., Edenton St., and Salisbury St., respectively. Furthermore, define y to be the total cars per day exiting the intersection of Jones St. and Wilmington St. to the east along Jones St.

Solution

Let's begin by analyzing the given map and setting up the necessary constraints and equations for the traffic flow system.

(a) Identify Constraints and Set Up the Linear System:

i. Constraints:

We will consider traffic conservation principles at each intersection: the sum of cars entering an intersection should equal the sum of cars leaving.

Here are the intersections with their respective constraints:

  1. Intersection at Jones St. and Salisbury St. (Top Left):

    • Incoming cars: 4300 (north on Salisbury St.) + xx (east on Jones St.) = Outgoing cars: 1200 (west on Jones St.) + zz (south on Salisbury St.)
    • Equation: 4300+x=1200+z(Eq. 1)4300 + x = 1200 + z \quad \text{(Eq. 1)}

  2. Intersection at Jones St. and Wilmington St. (Top Right):

    • Incoming cars: 5100 (west on Edenton St.) + xx (west on Jones St.) = Outgoing cars: 2300 (north on Wilmington St.) + yy (east on Jones St.)
    • Equation: 5100+x=2300+y(Eq. 2)5100 + x = 2300 + y \quad \text{(Eq. 2)}

  3. Intersection at Edenton St. and Wilmington St. (Bottom Right):

    • Incoming cars: 3800 (north on Wilmington St.) + uu (north on Wilmington St.) = Outgoing cars: 5100 (west on Edenton St.) + 2300 (north on Wilmington St.)
    • Equation: 3800+u=5100+2300(Eq. 3)3800 + u = 5100 + 2300 \quad \text{(Eq. 3)}

  4. Intersection at Edenton St. and Salisbury St. (Bottom Left):

    • Incoming cars: 4100 (east on Edenton St.) + vv (east on Edenton St.) = Outgoing cars: 6600 (south on Salisbury St.) + zz (north on Salisbury St.)
    • Equation: 4100+v=6600+z(Eq. 4)4100 + v = 6600 + z \quad \text{(Eq. 4)}

  5. Intersection at Wilmington St. and Edenton St. (Middle):

    • Incoming cars: yy (east on Jones St.) + 4300 (north on Salisbury St.) = Outgoing cars: (

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Traffic Flow Modeling

Formulas

Conservation of cars at each intersection: Incoming = Outgoing
Linear Equations derived from traffic flow constraints

Theorems

Gauss-Jordan Elimination
Conservation Laws

Suitable Grade Level

Undergraduate Level

Related Recommendation

Traffic Flow Analysis using Linear Equations in Downtown Raleigh

Solve Traffic Flow System with Linear Equations for Intersections A, B, C, D

Solving Traffic Signal Cycles Using Systems of Linear Equations

Traffic Flow Optimization on a One-Way Street System

Traffic Flow Problem Solved Using Flow Balance Equations