Question.2930 - 1. Determine how far a fire truck would travel from each station to every Regional Demand Point. Hint: Assume city streets are laid out in a rectangular grid and vehicles always travel in a North-South or East-West direction (there are no angled streets). 2. If you know the distance that a fire truck travels, how can you predict the time required to travel that distance? Develop a method to predict the time required for any distance that a fire truck might travel. Hint: Use the sample runs in Figure 2: One such predictive expression would take the form: 3.Make a table showing the response time for every Regional Demand Point from every station. Which Regional Demand Points are not covered by any existing station within five minutes? Within six minutes? Within seven minutes 4. What is the minimum number of stations that can cover all Regional Demand Points within four minutes? Within five minutes? Within six minutes? Within seven minutes?
Answer Below:
Questions 1. Determine how far a fire truck would travel from each station to every Regional Demand Point. Hint: Assume city streets are laid out in a rectangular grid and vehicles always travel in a North-South or East-West direction (there are no angled streets). Solution: Given that the city streets are laid out in a rectangular grid. So, the sum of the absolute value of the difference between the and coordinates of stations and Regional Demand Point will given the distance between the specific station and the RDP. For example: For Station 1: X = 1 and Y = 2 RDP 1: X = 0.7 and Y = 3.6 Distance between Station 1 and RDP 1 = abs(1 – 0.7) + abs(2 – 3.6) = 1.3 Since, distance cannot be negative, we take the absolute value. So, the distance between Station 1 and RDP 1 is 1.3 Continuing in the similar way the distance between each station and RDP is given by, Station 1 Station 2 Station 3 Station 4 Station 5 Station 6 RDP 1 1.9 1.6 2.6 5.1 3.8 2.8 RDP 2 6 3.1 2.1 3.8 1.3 3.9 RDP 3 3.9 1 0.4 4.7 2.8 2.4 RDP 4 0.6 2.7 3.7 2.6 4.9 1.9 RDP 5 3.7 2.6 1.8 2.5 1.4 1.6 RDP 6 4.6 6.5 5.7 1.4 3.3 3.7 RDP 7 1.3 4.2 5.2 3.5 6.4 3.4 RDP 8 5.9 3 2 3.7 1 3.8 RDP 9 5.7 3.2 2.4 3.5 0.6 3.6 RDP 10 4.6 3.9 3.1 2.4 0.7 2.5 RDP 11 4.3 6.2 5.4 1.1 3.8 3.4 RDP 12 3.2 5.1 4.3 2.2 5.1 2.3 RDP 13 3.6 5.1 4.3 1.4 1.9 2.3 RDP 14 1.3 4 5 3.3 6.2 3.2 RDP 15 3.7 5.6 4.8 1.1 2.4 2.8 RDP 16 0.3 2.6 3.6 3.1 4.8 1.8 RDP 17 4.5 6.4 5.6 1.3 3.2 3.6 2. If you know the distance that a fire truck travels, how can you predict the time required to travel that distance? Develop a method to predict the time required for any distance that a fire truck might travel. Hint: Use the sample runs in Figure 2: One such predictive expression would take the form: TIME (MINUTES) = CONST + COEFFICIENT x DISTANCE (MILES) Once you have developed such an expression from the available data, is your solution a valid expression for the time? If so,why? If not, why not? Are other expressions more valid? Explain and develop, if applicable. Solution: Let us use the above method to find the distance between the specified station and the data. So, we have, For example, given that Station 2: X = 1.5 and Y = 4.4 Add-X = 1.2 and Add-Y = 3.8 Distance = abs(1.5 – 1.2) + abs(4.4 – 3.8) = 0.9 Thus distance between the first address and station 2 is 0.9. Thus, distance between the addresses and the stations are as follows. Station Station (X) Station (Y) Add-X Add-Y Distance 2 1.5 4.4 1.2 3.8 0.9 1 1 2 1.3 4.3 2.6 4 3.7 1.5 2.6 2.3 1.9 5 4.2 3.9 3.8 1.8 2.5 4 3.7 1.5 4 0.8 1 6 2.5 2.6 1.2 3.1 1.8 3 2.4 4.5 1.5 2.6 2.8 3 2.4 4.5 4 4.2 1.9 6 2.5 2.6 1.7 4.1 2.3 1 1 2 1.8 1.1 1.7 2 1.5 4.4 0.2 4.2 1.5 2 1.5 4.4 0.3 4 1.6 1 1 2 0.8 2.3 0.5 5 4.2 3.9 4.9 1.8 2.8 3 2.4 4.5 0.4 4.4 2.1 4 3.7 1.5 4.1 0.1 1.8 1 1 2 0.1 1.6 1.3 1 1 2 4 1.5 3.5 6 2.5 2.6 2.9 3.9 1.7 1 1 2 1.5 4.9 3.4 1 1 2 1.8 3.8 2.6 4 3.7 1.5 1.9 1.1 2.2 4 3.7 1.5 4.9 3 2.7 4 3.7 1.5 4.8 0.2 2.4 3 2.4 4.5 4 4.7 1.8 1 1 2 1.1 0.6 1.5 4 3.7 1.5 3.9 0.3 1.4 4 3.7 1.5 4.4 0.9 1.3 3 2.4 4.5 4.4 3.6 2.9 2 1.5 4.4 2.4 2.2 3.1 Now, we can use line of best fit to derive an expression. For this let us plot the data by taking the distance travelled in X axis and the time taken in Y axis. The data for the X and Y axis is, Distance (d) Time (t) 0.9 3.4 2.6 8.2 1.9 4.6 2.5 7.8 1 3.4 1.8 5.9 2.8 9.2 1.9 5.9 2.3 7.8 1.7 4.4 1.5 4.5 1.6 5.5 0.5 1.7 2.8 8.9 2.1 6.8 1.8 4.9 1.3 5.3 3.5 11.1 1.7 4.9 3.4 11.7 2.6 8.2 2.2 7.9 2.7 8.3 2.4 7.2 1.8 4.7 1.5 4.6 1.4 4.7 1.3 4.7 2.9 8.3 3.1 9.3 The graph and the line of best fit are given in the following graph. 00.511.522.533.54 0 2 4 6 8 10 12 14 Time Time Linear(Time) Now, the equation is It could be checked that all the points lie close to the line of best fit. In the above equation, represents time in minutes and is distance in miles. So, this equation fits the data best. So, the expression is valid for time. 3. Make a table showing the response time for every Regional Demand Point from every station. Which Regional Demand Points are not covered by any existing station within five minutes? Within six minutes? Within seven minutes? Solution: In order to determine the response time for every RDP, let us use the linear equation found in part (2). The linear equation is, The following equation gives the response time for every RDP from every station. RDP Station 1 Station 2 Station 3 Station 4 Station 5 Station 6 d Response time t d Response time t d Response time t d Response time t d Response time t d Response time t 1 1.9 5.99658 1.6 5.06982 2.6 8.15902 5.1 15.88202 3.8 11.86606 2.8 8.77686 2 6 18.6623 3.1 9.70362 2.1 6.61442 3.8 11.86606 1.3 4.14306 3.9 12.17498 3 3.9 12.17498 1 3.2163 0.4 1.36278 4.7 14.64634 2.8 8.77686 2.4 7.54118 4 0.6 1.98062 2.7 8.46794 3.7 11.55714 2.6 8.15902 4.9 15.26418 1.9 5.99658 5 3.7 11.55714 2.6 8.15902 1.8 5.68766 2.5 7.8501 1.4 4.45198 1.6 5.06982 6 4.6 14.33742 6.5 20.2069 5.7 17.73554 1.4 4.45198 3.3 10.32146 3.7 11.55714 7 1.3 4.14306 4.2 13.10174 5.2 16.19094 3.5 10.9393 6.4 19.89798 3.4 10.63038 8 5.9 18.35338 3 9.3947 2 6.3055 3.7 11.55714 1 3.2163 3.8 11.86606 9 5.7 17.73554 3.2 10.01254 2.4 7.54118 3.5 10.9393 0.6 1.98062 3.6 11.24822 10 4.6 14.33742 3.9 12.17498 3.1 9.70362 2.4 7.54118 0.7 2.28954 2.5 7.8501 11 4.3 13.41066 6.2 19.28014 5.4 16.80878 1.1 3.52522 3.8 11.86606 3.4 10.63038 12 3.2 10.01254 5.1 15.88202 4.3 13.41066 2.2 6.92334 5.1 15.88202 2.3 7.23226 13 3.6 11.24822 5.1 15.88202 4.3 13.41066 1.4 4.45198 1.9 5.99658 2.3 7.23226 14 1.3 4.14306 4 12.4839 5 15.5731 3.3 10.32146 6.2 19.28014 3.2 10.01254 15 3.7 11.55714 5.6 17.42662 4.8 14.95526 1.1 3.52522 2.4 7.54118 2.8 8.77686 16 0.3 1.05386 2.6 8.15902 3.6 11.24822 3.1 9.70362 4.8 14.95526 1.8 5.68766 17 4.5 14.0285 6.4 19.89798 5.6 17.42662 1.3 4.14306 3.2 10.01254 3.6 11.24822 From the above data we can say the following. RDPs that are not covered within 5 minutes from any one of the stations are RDP 1, RDP 12 RDP that are not covered within 6 minutes from any one of the stations is RDP 12 All the RDPs are connected from any one of the stations within 7 minutes. 4. What is the minimum number of stations that can cover all Regional Demand Points within four minutes? Within five minutes? Within six minutes? Within seven minutes? Solution: Checking Part (3), we can infer that the existing 6 stations cannot cover all Regional demand points within four minutes, five minutes and six minutes. Now, let us check the minimum number of stations that can cover all the RDPs within seven minutes. The following table enumerates the stations that can cover each RDPs that can cover within 7 minutes. RDP Stations 1 2, 1 2 5, 3 3 3, 2 4 1, 6 5 5, 6 6 4 7 1 8 5, 3 9 5 10 5 11 4 12 4 13 4, 5 14 1 15 4 16 1, 6 17 4 In seven minutes the RDPs 6, 7, 9, 10, 11, 12, 14, 15, and 17 can be covered by the stations, Station 1, Station 4, and Station 5. These Stations also cover RDPs 1, 2, 4, 5, 8, 13, and 16. The only RDP left is RDP 3. Stations 3 and 2 will connect it. Thus, All the RDPs can be covered with just 4 Stations namely Station 1, Station 2, Station 4 and Station 5 within seven minutes. Therefore the minimum number of Stations that could connect all the RDPs within seven minutes is four.More Articles From Maths
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