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A township in Guilin City has a total of 2,000 acres of arable land, including 400 acres of sandy land, 600 acres of clay land, and 1,000 acres of neutral land. The second category is vegetables; The third category is cash crops, represented by the local specialty - honeysuckle.

Table 1 Statement of costs and benefits for the cultivation of various types of crops.

Crop type Cost (Yuan mu) Income (Yuan Mu).

Sandy land Clayey land General land.

Rice 200 160 150 300

Vegetables 300 290 280 500

Honeysuckle 260 260 240 450

In order to prevent the tendency of monoculture of various crops, on the basis of ensuring that the whole township has sufficient food rations, the cultivation of various crops is developed in a coordinated manner. According to the planting situation and benefits in previous years, the township proposed the proportion of the sown area of three crops, with rice area and vegetable area

Honeysuckle area = 2:1:1 is appropriate.

According to the total planting area of 2,000 mu of land in the township, the maximum planting area of rice, vegetables and honeysuckle can be set to 1,000 mu, 500 mu and 500 mu respectively. Find the optimal planting plan. There needs to be a detailed question process, thank you.

Mathematical modeling is mainly about developing thinking, and there is no fixed answer to the problem. I advise you to use your brain with your peers and don't look for so-called answers, because then you won't experience the unique charm and fun of mathematical modeling.

Question A: Digital camera positioning.

Digital camera positioning has a wide range of applications in traffic supervision (e-police) and other aspects. The so-called digital camera positioning refers to the use of digital cameras to take photos of objects and determine the position of certain landmark points on the surface of the object. The most commonly used positioning method is binocular positioning, which uses two cameras to locate the position.

For a feature point on an object, the image of the object is captured by two cameras fixed in different positions, and the coordinates of the point on the image plane of the two cameras are obtained respectively. As long as the exact relative position of the two cameras is known, the coordinates of the feature point in the coordinate system of the fixed camera can be obtained by geometric methods, that is, the position of the feature point is determined. Therefore, for binocular positioning, it is crucial to accurately determine the relative position of the two cameras, a process called system calibration.

One of the methods of calibration is: draw a number of points on a flat plate, take pictures with these two cameras at the same time, obtain the image points of these points on their image planes respectively, and use the geometric relationship of these two groups of image points to obtain the relative positions of these two cameras. However, we cannot directly obtain a "point" without geometric dimensions on either the object plane or the image plane.

The actual practice is to draw a number of circles (called targets) on the plane of objects, and their centers are geometric points. Their images are generally distorted, as shown in Figure 1, so the image of the center of the circle must be accurately found from the images of these circles on the target, and calibration can be achieved.

Figure 1 Image of a circle on a target.

Someone designed the target as follows, take a square with a side length of 100mm, take four vertices (corresponding to a, c, d, e) as the center of the circle, and 12mm as the radius as the circle. Take B at a distance of 30 mm from point A on the edge of the AC as the center of the circle, and 12 mm as the radius to make a circle, as shown in Figure 2.

Figure 2 Schematic diagram of the target.

Figure 3 Image of the target.

1) Establish mathematical models and algorithms to determine the image coordinates of the center of the circle on the target in the image plane of the camera, where the origin of the coordinate system is taken as the focal point of the camera, and the x-y plane is parallel to the image plane;

2) for the target and its image given by Fig. 2 and Fig. 3 respectively, calculate the image coordinates of the center of the circle on the target on the image plane, the image distance of the camera (i.e., the distance from the focal point to the image plane) is 1577 pixel units (1 mm is about one pixel unit), and the camera resolution is 1024 786;

3) Design a method to test your model and discuss the accuracy and stability of the method;

4) Establish a mathematical model and method for giving the relative positions of two fixed cameras with this target.