Solution: If the price of 10a is increased on the basis of 59 yuan, 10a will be sold less.
According to the meaning of the question
(50+ 10a-30)×(300- 10a)= 8700
(20+ 10a)×(30-a)=870
(a+2)(a-30)=-87
Answer? -28a+27=0
(a- 1)(a-27)=0
A= 1 or a=27
When a= 1, the price increases by 10 yuan, and 300- 10× 1=290 pieces are sold.
When a=27, the price will increase by 27× 10=270 yuan, and 300- 10×27=30 pieces will be sold (this price is unrealistic).
Belonging to theoretical calculation
2. A company produces a certain commodity. The cost of each product is 3 yuan and the price is 4 yuan. The annual sales volume is 654.38+ten thousand pieces. In order to cope with the global economic crisis in 2009, the company is going to spend some money on advertising. According to experience, the annual advertising investment is X (10,000 yuan), and the product sales will be Y times of the original, while
y=-x？ /10+7/10x+7/10IF: annual profit = total sales-cost-advertising fee.
(1) Can the company's annual profit reach 1.5 million? Can it reach 1.6 million?
(2) Can the company's annual profit reach 6.5438+0.7 million? If yes, please calculate how much the advertisement should be at this time. If not; Please provide a justification for the answer.
Solution: Suppose the annual profit is one million yuan,
a=4× 10y-3× 10y-x
=40y-30y-x= 10y-x
= 10×(-x？ / 10+7/ 10x+7/ 10)-x
=-x？ +7x+7x
=-x？ +6x+7
When a= 15
-x？ +6x+7= 15
x？ -6x+8=0
(x-2)(x-4)=0
X=2 or 4
When the advertising fee is 20,000 yuan or 40,000 yuan, the profit will reach 6.5438+0.5 million yuan.
When a= 16
-x？ +6x+7= 16
x？ -6x+9=0
(x-3)？ =0
x 1=x2=3
When the advertising fee is 30,000 yuan, the profit will reach 6.5438+0.6 million yuan.
When a= 17
-x？ +6x+7= 17
x？ -6x+ 10=0
Discriminant = 36-40 =-4
So the profit can't reach 1.7 million.
3. Several people in an interest group send each other a greeting card during the New Year. It is known that the whole group has issued a 132 card. Please find out the number of people in this group.
Solution: Suppose there is a person in this group.
According to the meaning of the question
a×(a- 1)= 132
Answer? -a- 132=0
(a- 12)(a+ 1 1)= 0
A= 12 or a=- 1 1 (omitted)
There were 12 people, and each person received12-1=11card.
4. Party A and Party B complete a project in 6 days, and it is known that it will take 5 days for Party A to do it alone. How many days does it take for Party A to complete it alone?
Solution: Suppose it takes X days for B to finish it alone.
6× 1/x+6× 1/(x+5)= 1
6x+30+6x=x？ +5 times
x？ -7x-30=0
(x- 10)(x+3)=0
X= 10 or x=-3 (omitted)
B it takes 10 days to complete it alone.
A it takes 10+5= 15 days to complete it alone.
5. A village plans to build a rectangular vegetable greenhouse as shown in the figure, and the length-width ratio is required to be 2: 1. In the greenhouse, an open space with a width of 3M is reserved along the front inner wall, and passages with a width of 1M are reserved on the other three inner walls. When the length and width of rectangular greenhouse are respectively what, the vegetable planting area is 288m2?
Solution: If the width is one meter, then the length is 2 meters.
According to the meaning of the question
(2a-3- 1)(a- 1- 1)= 288
(2a-4)(a-2)=288
(a-2)？ = 144
a-2= 12
a=2 12
A= 14 or a=- 10 (irrelevant, omitted)
Therefore, when the width is 14 m and the length is 28 m, the area of vegetable planting area is 288 square meters.
6. A village plans to build a channel with an isosceles trapezoid cross section with an area of 10.5m? The upper bottom surface is 3m wider and 2m deeper than the lower bottom surface. How wide should the upper bottom be dug?
Solution: If the upper bottom is one meter, the lower bottom is a-3 meters, and the depth is a-2 meters.
According to the meaning of the question
(a+a-3)×(a-2)/2= 10.5
(2a-3)(a-2)=2 1
2a？ -5a- 15=0
(2a+3)(a-5)=0
A=5 or a=-2/3 (irrelevant, omitted)
So the upper bottom is 5 meters.
7. A shop sells a batch of shirts. If each shirt is profitable in 40 yuan, it can sell 20 shirts a day. In order to reduce inventory and increase profits as soon as possible, the mall decided to cut prices. If you reduce each shirt by 1 yuan, you can sell 2 more shirts a day on average. Q: When each shirt is reduced, the average daily profit will be 1200 yuan?
Solution: Suppose the price is reduced by one yuan, then a lot of 2a ones are sold.
(40-a)×(20+2a)= 1200
800-20a+80a-2a？ = 1200
Answer? -30a+200=0
(a- 10)(a-20)=0
A= 10 or a=20.
That is to say, it is enough to reduce the price by 10 yuan or 20 yuan.
8. The average monthly output growth rate of a factory in the first quarter is X, the output value in January is A yuan, and the output value in March becomes1.21a. What is the value of x?
Solution: let the increment of yield be x.
a( 1+x)？ = 1.2 1a
( 1+x)？ = 1. 1
1+x= 1. 1 or 1+x=- 1. 1.
X=0. 1 or -2. 1 irrelevant, omitted.
Growth rate = 10%
9. To manufacture a product, because the cost is reduced by 36% twice in a row, what is the average percentage of cost reduction each time?
Solution: let the cost be a and reduce x every time.
a( 1-x)？ =a×( 1-36%)
( 1-x)？ =0.64
1-x=0.8 or 1-x=-0.8.
X=0.2 or 1.8 (irrelevant, omitted)
Reduce by 20%
10. A store buys a batch of goods at the price of 2 1 yuan each. Commodities can be priced by themselves. If each product is one yuan, you can sell (350- 10a) pieces. However, the price bureau limits the profit of each product to no more than 20%. How many pieces does the store need to purchase to make a profit in 400 yuan? What's the price of each piece?
Solution: according to the meaning of the problem
(a-2 1)(350- 10a)=400
350a-7350- 10a？ +2 10a=400
Answer? -56a+775=0
(a-25)(a-3 1)=0
A=25 or a=3 1
Because the profit does not exceed 20%, a is at most 2 1×( 1+20%)=25.2.
So a=3 1 is irrelevant, so it is omitted.
So a=25
The price is 25 yuan, and the stock is 350- 10×25= 100.
1 1. If the number of people in a travel agency does not exceed 25, the per capita cost is 1000 yuan. If there are more than 25 people, the per capita travel expenses will be reduced in 20 yuan, but the per capita expenses shall not be lower than the charging standard in 700 yuan. The employees of a certain unit paid 27,000 yuan for a tour. How many people participated in the tour?
Solution: First, judge.
There are more than 25 people in this unit.
Because if there are 25 people, then the amount of money used is 25 ×1000 = 25,000 yuan.
So more than 25 people
Suppose one person is added, and the per capita cost is 1000-20a-20 yuan.
( 1000-20a)×(25+a)=27000
25000-500a+ 1000a-20a？ =27000
20a？ -500 Amps +2000=0
Answer? -25a+ 100=0
(a-5)(a-20)=0
A=5 or 20
When a=20, the per capita cost = 1000-20× 20 = 600.
So a=20 doesn't matter, so I gave up.
So 25+5=30 people travel.
12. Use a 20-meter-long iron wire to form a rectangle with an area of 25 square meters. What is the length of this rectangle?
Solution: If the length is x meters and the width is 20/2-x= 10-x meters.
According to the meaning of the question
( 10-x)x=25
x？ - 10x+25=0
(x-5)？ =0
x 1=x2=5
So the length = width = 5m of a rectangle is a square.
13, a school-run factory 10 produced 200 sets of A products. By improving the production technology, the total output in February and March increased by the same percentage point compared with the previous month, so the total output in the first quarter reached 1400 units. What is the percentage?
Solution: let this percentage be a.
200+200( 1+a)+200( 1+a)？ = 1400
Let 1+a=t
t？ +t-6=0
(t-2)(t+3)=0
T=2 or t=-3 (truncation)
So 1+a=2.
a= 1= 100%
14, with two numbers. Their sum is 13, and their product is -48. What are these two numbers?
Solution: Let one number be a and the other number be13-a.
a( 13-a)=-48
Answer? - 13a-48=0
(a- 16)(a+3)=0
A=-3 or a= 16
When a=-3, another number is 16.
When a= 16, another number is -3.
15. In order to expand the 1 swimming pool with a length of 200m and a width of 60m into a large water amusement park with a circumference of 600m, the length of the swimming pool is increased by xm. So when x =, the area of water swimming pool is 20000 square meters. If we can find the value of x? If you can't explain why.
Solution: after the length is increased, it is 100+x meters.
At this time, the width is (600/2- 100-x)=200-x meters.
( 100+x)(200-x)=20000
20000+200x- 100x-x？ =20000
x？ - 100x=0
x(x- 100)=0
X= 100 or x=0 (omitted)
The length is increased by 100m and the width is increased by 200- 100-60 = 40m.
16, Class 1, Grade 9, Group 2 At a party, each student presented a photo to the other students. This activity sent out 90 photos. How many students are there in this group?
Solution: There are X students.
Each student received x- 1 photo.
A * * * has x students.
Then the total number = number of students × number of photos received by each student.
The formula x(x- 1)=90.
x？ -x-90=0
(x- 10)(x+9)=0
X= 10 or -9 (truncated)
So there are 10 students.
17 As shown in the figure, use 12m long wood to make a herringbone window with a crosspiece in the middle.
(1) If the light penetrating into the window reaches 4.5m? What is the length and width of the window at this time?
(2) If the light penetrating into the window reaches 6m? What is the length and width of the window at this time?
(3) If the light penetrating into the window reaches 7m? Is it possible? Why?
Solution: (1) Let the length be a meter and the width be b meter.
According to the meaning of the question
2a+3b= 12( 1)
ab=4.5(2)
By (1)
2a= 12-3b
By (2)
2ab=9
( 12-3b)b=9
4b-b？ =3
b？ -4b+3=0
(b- 1)(b-3)=0
B= 1 or b=3.
When b= 1, A=4.5.
When b=3, a= 1.5.
(2)
2a+3b= 12
ab=6
The process of solution is omitted.
a=3
b=2
(3)
2a+3b= 12
ab=7
2ab= 14
( 12-3b)×b= 14
3b？ - 12b+ 14=0
Discriminant144-12×14 =-24.
So it is impossible to reach 7m? .
18, a pile of coal, using 40% of the total, brought in 24 tons. At this time, the tonnage is two-thirds of the original. How many tons is this pile of coal?
Suppose there was a ton.
a×( 1-40%)+24=2/3a
a-0.4a+24=2/3a
2/3a-3/5a=24
1/ 15a=24
A=360 tons