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15.3 Linear Programming 线性规划
(1)
Mr. Tan planned to buy durian and watermelon according to the following conditions.

I The amount of watermelon that was bought exceed the amount of durian that was bought at least 5.

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.

III The amount of watermelon that was bought cannot be more than two time of the amount of durian that was bought.

Using the variables x and y to represent the amount of durian and watermelon respectively, write out the inequalities that satisfy the given conditions.

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.

(a) Find the range of the amount of watermelon that could be bought if 14 durians were bought.

(b) Find the maximum amount of fruits that could be bought by Mr. Tan.

, ,

(a)

(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:

I The total planted area cannot more than 70 acre.

II The farmer has modal RM9600 only.

III The store can only places not more than 3600 gunnies of harvest.

(a) Write an inequality for each of the given condition.

(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.

(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.

(d) If 20 acre of corn is planted, find the range of the amount of paddy in acre that could be planted.

(a) ,,

(c) 40, 20, RM88

(d)

(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.

I At least 20 pieces of Malay costumes were bought

II At least 40 pieces of the sum of Indian costumes and Malay costumes were bought.

III The amount of Malay costumes were bought exceed 3 times of the amount of Indian costumes were bought not more than 15 pieces

IV The total expenses for buying both of the clothes not more than 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.

(a) Find the range of the amount of Malay costumes that could be bought if 20 pieces of Indian costumes were bought.

(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.

y 20, x + y 40, y 3x + 15, 80x + 60y 4800

(a) 20 y 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,

(a) write the inequalities other than (x ≥ 0, y ≥ 0) that satisfy the conditions above，

(b) using 1cm to 1 unit for both axes, shad and label with R, the region that satisfies the conditions above，

(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.