production plan

A factory produces three kinds of standard parts, A, B and C, and its profits are 3 1.5 and 2 yuan respectively. If the factory only produces one kind of standard parts, it can produce 800, 1200, 1000 pieces of standard parts A every day, but standard parts A need some special treatment, with a maximum of 600 pieces per day. At least 200 B standard parts are produced every day.

(1) How should the factory arrange the production plan to maximize the daily profit? Try to establish a general mathematical model;

(2)

As an example, the solution of this problem is obtained.

Question b

Tree planting problem

A group of 6 boys and 5 girls are going to plant trees on Sunday. According to past experience, boys dig 20 holes on average every day, or plant 30 trees, or water 25 planted saplings; On average, girls dig 10 holes every day, or plant 20 trees, or water 15 seedlings.

(1) Try to establish a general mathematical model, which can reasonably arrange and organize the trees planted by manpower.

(2) Take an example to find the solution of this problem.

C theme

Transition curve problem of train curve

When the train runs on a curve, according to the mechanical principle, centrifugal force F will be generated. At the junction of straight track and curve (arc), the centrifugal force on the train will suddenly change from zero to f, thus damaging the line and vehicles, making passengers feel uncomfortable and even dangerous. For this reason, the train track adopts the method of "the outer rail is super high" at the bend, that is, the outer rail on the bend is raised to a certain height, so that the train tilts, and the centripetal force generated in this way offsets part of the centrifugal force to ensure the safe operation of the train. In order to make the straight track with the same height smoothly connect with the arc of the outer rail superelevation and avoid the sudden appearance of centrifugal force, a curve should be added between the curve and the straight track, so that the centrifugal force on the train increases evenly from zero to f, and the superelevation of the outer rail increases gradually from zero to h, and the added curve is called a transition curve.

There is an existing railway bend with the original turning radius R=400m, which is suitable for train speed.

120km∕h。 As the train speeds up, it is required to change this curve to adapt to the train speed of 200 per hour.

Km∕h, and the original length is required to be 200.

The transition curve of m is transformed together. Try to discuss the following questions: