(1) Find the functional relationship between the daily average sales volume y (box) and the sales price x (yuan/box).
(2) Find the functional relationship between the daily average sales profit w (yuan) of wholesalers and the sales price x (yuan/box).
(3) When the price of each box of apples is several yuan, the maximum profit can be obtained? What is the maximum profit?
Test center: Select the function type according to the actual problem.
Special topic: application questions; Properties and applications of functions.
Analysis: (1) According to the selling price of each box in 50 yuan, 90 boxes are sold every day on average. If the price rises by 1 yuan, and three boxes are sold on average every day, the functional relationship between the average daily sales y (box) and the sales price x (yuan/box) can be obtained.
(2) The average daily sales profit of wholesalers = average daily sales volume × profit per box, and a conclusion can be drawn;
(3) Using collocation method, combined with monotonicity of function, we can get a conclusion.
Solution: (1) According to the selling price of each box in 50 yuan, 90 boxes are sold every day on average. Every time the price rises 1 yuan, 3 boxes are sold on average every day.
Available daily average sales y=90-3(x-50),
Simplification: y=-3x+240, (50 ≤ x < 55);
(2) The average daily sales profit of this wholesaler w = (x-40) (-3x+240) =-3x2+360x-9600 (0 < x ≤ 55);
(3)w =-3 x2+360 x-9600 =-3(x-60)2+ 1200
∵ 0 < x ≤ 55, ∴ function monotonically increases on (0,55),
∴ when x=55 yuan, the maximum value of w is 1 125 yuan.
When the price of each box of apples is 55 yuan, the maximum profit can reach 1 125 yuan.
Comments: This question examines the construction of function model and students' ability to solve practical problems by using mathematical knowledge, which is an intermediate question.
Please adopt it.