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# 第二次模拟赛赛题

A: The Irrigation Problem

Design a linear irrigation system to uniformly spray water on a rectangular field if the irrigation device moves across the field at a constant rate in a linear fashion as shown in

Figure 1.

Figure 1 Linear Irrigation System.

The linear irrigation system consists of a long water pipe, generally set on wheels that keep it above the level of the plants. Nozzles are placed periodically alone the pipe, and each sprays water in a circular region. The entire system moves slowly down the field, watering the plants beneath as it moves. You have 100 feet of pipe and 20 nozzles available. You have enough pumps to keep up the pressure on each of the nozzles so they deliver a uniform spray to a circular region 50 feet in radius. How far apart should the nozzles be placed to produce the most uniform distribution of water on a field 1000 feet wide?

B: Predicting Pumpkin Weights

A typical contest in the fall is to guess the weight of a pumpkin without a scale. Using just a few measurements, can you predict the weight of a pumpkin? Your assignment is to write a paper that uses the modeling process to answer this question.

First, you will want to discuss variables and assumptions such as geometric similarity. You will also have to use one or more (length) measurements to estimate weight. If you have access to some pumpkins, you may wish to decide which variables are important and take your own measurements. If not, you may use the data found below.

After you have your data, solve/verify the model. You may want to use more than one technique, using one or more measurements. Which model did you feel was the "best"? How accurate is your model? Discuss the balance between accuracy and simplicity. How will measurement error affect your predictions? Overall, what model do you choose to predict a pumpkin's weight and why?

What about implementation? If you're just trying to predict one pumpkin's weight, then a complicated formula might be okay. However, suppose an organization is selling pumpkins to the general public and wants to base its prices on weight. If a scale is unavailable, then how would you suggest they implement your model?

If available, test your model on some new pumpkins. Discuss your conclusions.

2003 Data

circumferences height horiz. vertical 1 vertical 2 weight 4 22.8 19.3 19.2 0.25 6 30.4 26.4 26.5 0.6 7.5 31.1 28.7 29 0.75 9.7 50 42.5 42.3 2.4 14.7 46 48.8 48.8 2.8 19.7 66 66.5 66 7 31.2 66 85 88 10.1 24.7 85 86 81 12.5 32.7 83.7 97.5 99.5 16.9 23.2 101.2 93.3 89.8 17.2 all in cm accurate to +/- 0.1lbs

2004 Data height horiz. 7.4 27.8 8.1 41 8.4 50.5 12.5 47.5 18 67.5 21 74.6 19 85.8 26.5 78.3 25.4 90.1 23.5 101.5 all in cm

circumferences vertical 1 vertical 2 27 27.5 35.5 35.8 41.7 42 46 46 64.5 66 73.5 72.5 81 79 90 89 89.5 89 93 94.5

weight 0.625 1.125 2.125 2.75 7.625 8.875 11.25 12.75 16 19.375 accurate to within 1/8 lb.

Note: The 2004 data includes a “white” pumpkin. The rest are the typical orange

variety. Can you determine which is the “white” one?