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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

15.3 线性规划 Linear Programming
15.3 Linear Programming 线性规划
(1)
Mr. Tan planned to buy durian and watermelon according to the following conditions.
陈先生计划依据下列的条件购买榴莲和西瓜

I The amount of watermelon that was bought exceed the amount of durian that was bought at least 5.
所购买的西瓜数目必须超过所购买的榴莲数目至少5粒。

II The price of a durian is RM50 each and the price of a watermelon is RM40 each. Mr. Tan is able to spend RM2000 only.
每粒榴莲的价格是RM50和每粒西瓜的价格是RM40. 陈先生只能消费RM2000.

III The amount of watermelon that was bought cannot be more than two time of the amount of durian that was bought.
所购买的西瓜数目不可以超过所购买的榴莲数目。

Using the variables x and y to represent the amount of durian and watermelon respectively, write out the inequalities that satisfy the given conditions.
用变数x 和 y 分别表示榴莲和西瓜的数目,试写出满足以上条件的不等式

By using the scale of 2cm to 5 units at each axis, label and shad the region R that satisfies these inequalities. Use your graph to answer the following questions.
在两轴以2cm 对5单位的比例,并标示和涂阴影在满足以上不等式的区域R

(a) Find the range of the amount of watermelon that could be bought if 14 durians were bought.
若已经购买14粒榴莲,求还可以购买的西瓜数目的范围。

(b) Find the maximum amount of fruits that could be bought by Mr. Tan.
问陈先生最多可以购买多少粒水果。

Answer:

y \geq x + 5, 50x + 40y \leq 2000, y \leq 2x

(a) 19 \leq y \leq 28

(b) 46

(2)
A farmer plants corn and paddy in his farm. In a season, he plants x acre of corn and y acre of paddy. The farmer needs RM120 per acre to plant the corn and RM240 per acre to plant the paddy. The average harvest that the farm could get is 50 gunnies of corn per acre and 80 gunnies of paddy per acre. The farmer keeps all the harvest in a store before they are sold. The plantation plan for a season is limited by the following conditions:
一位农民在他的农场种植玉米和稻米。在一个季节,他种植 x 英亩玉米和 y 英亩稻米。农民每英亩需要 RM120 来种植玉米,每英亩需要 RM240 来种植稻米。农场可以获得的平均收成是每英亩 50 袋玉米和每英亩 80 袋稻米。农民在出售之前将所有收成保存在商店中。一个季节的种植计划受以下条件限制:

I The total planted area cannot more than 70 acre.
总种植不可超过70英亩

II The farmer has modal RM9600 only.
农民只有RM9600 的资金

III The store can only places not more than 3600 gunnies of harvest.
仓库只能存放不超过3600袋的收成

(a) Write an inequality for each of the given condition.
对以上各条件写出一个不等式。

(b) By using the scale of 2cm to 10 units for each axis, draw the graph of the inequalities. Label and shad the region R that satisfies the given conditions.
每个轴以2cm 对10单位的比例,画出那些不等式的图形。标示和涂阴影在满足以上各条件的区域R。

(c) If the net profit of each acre of corn is RM1.20 and each acre of paddy is RM2.00, determine how many acre of farm that should be used to plant corn and how many acre of farm should be used to plant paddy, so that the profit is the maximum. Find the maximum profit.
若每英亩玉米的净利润是RM1.20和每英亩稻米的净利润是RM2.00,计算应该种植多少英亩的玉米和种植多少英亩的稻米,使得所得到的利润是最高的。求最高利润。

(d) If 20 acre of corn is planted, find the range of the amount of paddy in acre that could be planted.
若已种植20英亩的玉米,求稻米还可以

Answer:

(a) x + y \leq 70,120x + 240y \leq 9600, 50x +80y \leq 3600

(c) 40, 20, RM88

(d) 0 \leq y \leq 30

(3)
A group of traditional dancers want to buy x pieces of Indian costumes for RM80 each and y pieces of Malay costumes for RM60 each with the following conditions.
一群的传统舞者依据下列的条件要买x件的印度服装每件RM80和y件马来服装每件RM60

I At least 20 pieces of Malay costumes were bought
至少购买20件马来服装

II At least 40 pieces of the sum of Indian costumes and Malay costumes were bought.
所购买的印度服装和马来服装的总数至少40件

III The amount of Malay costumes were bought exceed 3 times of the amount of Indian costumes were bought not more than 15 pieces
所购买的马来服装的数目要比所购买的印度服装的三倍多不超过15件

IV The total expenses for buying both of the clothes not more than RM4800.
买两种服装的总开销不超过RM4800

Write an inequality for each of the condition above. Hence, by using the scale of 2cm to 10 units for each axis, draw the graph for all the four inequalities. Please mark and shad the region R that satisfies the given conditions.
对以上各条件写出一个不等式。据此,在每个轴用2cm对10单位的比例,画出四个不等式。请标示和涂阴影在满足以上各条件的区域R。
Use your graph to answer the following questions.
用你的图来回答下列的问题

(a) Find the range of the amount of Malay costumes that could be bought if 20 pieces of Indian costumes were bought.
若已经买了20件印度服装,求还可以购买的马来服装件数的范围

(b) Set the maximum of the total amount of money that needed to spend for buying both kinds of costumes if the amount of Indian costumes and Malay costumes are the same.
若印度服装和马来服装的数目一样,定需要购买两种服装的消费总数的最大值。

Answer:
y \geq 20, x + y \geq 40, y \leq 3x + 15, 80x + 60y \leq 4800

(a) 20 \leq y \leq 53

(b) RM4760

(4)
A housewife has 3kg of flour and 1.75kg of butter to make two kinds of biscuits. For 100 pieces of biscuit A, she needs 240g of flour and 175g of butter. For 100 pieces of biscuit B, she needs 300g of flour and 100g of butter. The amount of biscuit B must not exceed 2 times of the amount of biscuit A. If each packet contains 100 pieces of biscuit, x and y represent the number of packet of biscuit A and B respectively,
一家庭主妇用3kg面粉和1.75kg奶油做两种饼干。 100块饼干A,她需要240克面粉和175克奶油。 100块饼干B,她需要300克面粉和100克奶油。饼干B的数量不得超过饼干A数量的2倍。如果每包含100块饼干, x和y分别代表饼干A和B的包数

(a) write the inequalities other than (x ≥ 0, y ≥ 0) that satisfy the conditions above,
写出除 (x ≥ 0, y ≥ 0) 以外满足上述条件的不等式

(b) using 1cm to 1 unit for both axes, shad and label with R, the region that satisfies the conditions above,
在两轴以1cm 对1单位的比例,涂阴影并以R标示满足以上条件的区域。

(c) find the maximum profit that could be made by the housewife if the profit from each packet of biscuit A and B is RM5 and RM8 respectively.
如果每包饼干 A 和 B 的利润分别为 RM5 和 RM8,求家庭主妇可以赚取的最大利润。

Answer:

(a) 240x + 300y \leq 3000, 175x + 100y \leq 1750

(c) RM 73