Mba-day25 best value problem - application problem

1. Train of thought - The formula of quadratic equation in one variable

y = ax^2 + bx + c

1. When a=0, The function is a linear equation , No maximum
2. When a<0 when , The parabolic opening is downward , The maximum value is (4ac-b^2)/4a, x = (x1+x2) /2
3. When a>0 when , The parabolic opening is upward , Only the minimum value ,x = -b/2a
4. Axis of symmetry （-b/2a, 4ac-b^2)/4a）

2. exercises

2.1 Example 1

The unit price of a batch purchased by a store is 20 Yuan of daily necessities , If at unit price 30 Yuan sales , Then it can be sold in half a month 400 Pieces of , Based on sales experience , Increasing the unit price will reduce the sales volume , That is, every increase in the sales unit price 1 element , The sales volume decreased accordingly 20 Pieces of . Improve （） Yuan , To get the maximum profit within half a month .

answer ： Improve （5） Yuan

 Explain ： Set to improve x individual 1 element , The sales volume decreases x individual 20 Pieces of 
 profits y = (30 + x - 20) * (400 - 20x)
= (10 + x)(400 - 20x), The parabolic opening is downward , There is a maximum 
 Make y = 0,  be x1 = -10, x2 = 20
x Maximum  =  (x1+x2) /2 = 5
 That is to improve （5） Yuan 

2.2 Example 2

A shopping mall sells a batch of famous brand shirts , On average, it can be sold every day 20 Pieces of , Every piece is profitable 40 element , To expand sales , Reduce inventory as soon as possible , The market decided to take appropriate measures to reduce prices , After investigation, it was found that , Every time the market reduces prices 1 element , On average, you can sell two more pieces per day , In order to maximize the average profit of the mall , The price should be reduced （） element

answer ： The price should be reduced （15） element

 Explain ： Set price reduction x individual 1 element , Sell more every day  2x  Pieces of 
y = (40 - x) * (20 + 2x) , The parabolic opening is downward , There is a maximum 
 Make y = 0,  be x1 = 40, x2 = -10
x Maximum  =  (x1+x2) /2 = 15

 The price should be reduced （15） element 

2.3 Example 3

The purchase price of a certain garment is 40 element , The price is per piece 50 element , It can be sold every month 200 Pieces of , If everything goes up 1 element , Less sales per month 10 Pieces of , If the mall wants to make the most profit , Then each piece should go up （） element

answer ： Every piece should go up （5） element

 Explain ： Set to rise x individual 1 element , Less sales  10x Pieces of 
y = (50 + x - 40) * (200 - 10x)
= (10 + x)* (200 - 10x),  The parabolic opening is downward , There is a maximum 
 Make y = 0,  be x1 = -10, x2 = 20
x Maximum  =  (x1+x2) /2 = 5

 Every piece should go up （5） element 

2.4 Example 4

Shopping malls sell certain goods , The purchase price is 100 element , When the selling price is 110 Yuan time , Can sell every day 100 individual . After research, we found that , Every rise in commodity prices 1 element , Daily sales decrease 2 Pieces of , Then the price is （） element , The total profit of this commodity is the largest

answer ： Pricing for （130） element , The total profit of this commodity is the largest

 Explain ： Set to rise x individual 1 element , Daily sales decrease 2x Pieces of 
y = (110 + x - 100) * (100 - 2x)
= (10 + x) * (100 - 2x),  The parabolic opening is downward , There is a maximum 
 Make y = 0,  be x1 = -10, x2 = 50
x Maximum  =  (x1+x2) /2 = 20
 pricing ：110 + 20 = 130

 answer ： Pricing for （130） element , The total profit of this commodity is the largest 

2.5 Example 5（ Wrong questions ）

The fixed cost of a manufacturing enterprise is 2000 element , Every product produced , Increase cost 60 element , Sales volume of products y And selling price x The relationship is y = 1000 - 10x, If the enterprise wants to make the most profit , The selling price of the product should be fixed （） element .

answer ： The selling price of the product should be fixed （80） element .

 Refer to the answer ：
 Set the product price x, Enterprise profit T = (x - 60)（1000 - 10x） - 2000
 Simplification T = -10x^2 + 1600x - 6200,  The parabolic opening is downward , There is a maximum 
x = -b/2a = -1600/(2*(-10)) = 80
 The selling price of the product should be fixed （80） element .


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 I didn't understand the meaning of the title ？
 Explain ： The price is x, 
 Set increase a A cost 60 element , Then the sales volume ：y = 1000 - 10x
y =  The price is  *  sales  = (x - 60 - 2000) * (1000 - 10x)