1. A factory has 360 kg of material A and 290 kg of material B, and plans to produce 50 pieces of products A and B with this material. It is known that the production of a product requires 9 kilograms of material A and 3 kilograms of material B, which will benefit 700 yuan; To produce a product B, 4kg of material A and10kg of material B are needed, and the profit is 1.200 yuan.
Q: (1) How many schemes are there to arrange the number of products A and B according to the above requirements?
(2) Let the total profit of products A and B be Y (yuan), and the number of pieces produced by one product be X. Try to write the functional relationship between Y and X, and use the functional properties to explain which production scheme has the largest total profit (1). What is the maximum profit?
Solution: (1) Assuming that X products A need to be produced, then (50-X) products B are obtained from the meaning of the problem:
9X + 4(50-X)《360 ……①
3X+ 10(50-X)《290 ……②
Solving Inequality Groups from ① x "32" and ② x "30 ∴ 30" x "32". Because X is a positive integer, X can take 30,365,438+0,32; The corresponding values of 50-X are 20, 19 and 18 respectively.
Scheme: 1, producing A30 pieces and B20 pieces; 2. Production of A3 1 piece and B 19 piece; 3. Production of 3.A32 pieces and B 18 pieces.
(2) If X pieces of product A are produced, product B (50-X) will be produced.
y = 700 X+ 1200(50-X)=-500 X+60000
Because x can only take 30,365,438+0,32; And y is a linear decreasing function, so when X=30, the value of y is the largest.
Ymin=-45000 (yuan)
2. At present, there are seven cards with the same back (but no front pattern can be seen from the back), and the front faces are printed with line segments, angles, side triangles, rectangles, regular pentagons, isosceles trapezoid and regular octagons respectively. If a card is randomly selected from these seven cards, the probability that the pattern on the selected card is both axisymmetric and centrally symmetric is ().
A.2/7 B.3/7 C.4/7 D.5/7 Answer B The straight lines, rectangles and regular octagons in these figures are consistent.
3. The parabola with the axis of symmetry x= 1 intersects with the X axis at points A and B, and intersects with the Y axis at point C, making a straight line BC, as shown in the figure. Point P is a moving point on line segment OB that does not coincide with points O and B. Point P is a parallel line of Y axis, intersection line BC is at point D, and intersection parabola is at point E, connecting CE and OD. Known OA: OB = 1: 3, tan∠DBP= 1. (1) search. (2) Find the maximum length of line segment DE; (3) Does the movement of point P have a certain moment? ① Will a quadrilateral with four vertices such as O, D, E and C become a parallelogram? ②△②△CDE and△△△ BDP are similar? If it exists, the coordinates of all points p that meet the conditions are requested; If it does not exist, please explain why.
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