y > 2x! 4 0 > 2(0)! 4

Transcription

1 y > 2x! 4 0 > 2(0)! 4? 0 >!4 y x

2 y! " 1 3 x + 3 y 6 0! x

3 y > 2x! 4 0 >? 2(0)! 4 0 >!4 y 6 y! " 1 3 x + 3 0! x

4 Linear Programming A School Board is investigating various ways of composing the faculty for a proposed new elementary school. They can hire teachers and aids. The amount of money the school district will have to spend on salaries each year depends on how many teachers and how many aids are hired. Let t = number of teachers hired. Let a = the number of aids hired. Let d = the number of thousands of dollars spent annually on faculty salaries. The board finds that the average teacher s annual salary is $35,000, and the average aide s salary is $25, Write an equation for the amount of money spent on salaries annually. d = 35t + 25a Therefore d is a function of two independent variables, t and a. The domain of this function is the set of all ordered pairs, (t, a) that fit the situation.

5 d = 35t + 25a Suppose that the Board finds the following requirements concerning the numbers of teachers and aides. i. The building can accommodate no more than 50 faculty members, total. ii. t + a 50 A minimum of 20 faculty members is needed to staff the school. t + a 20 iii. The school can not be run entirely by aides, so there must be at least 12 teachers. t 12 iv. For a proper teacher-to-aide ratio, the number of teachers must be at least half the number of aides. t (1/2)a v. It is impossible to hire a negative number of teachers or aides. t 0 and a 0

6 50 a t + a 50 t + a 20 t 12 t (1/2)a 2t a a 2t Feasible Region t

7 The school can spend at most 1 million dollars annually on salaries. Minimize the overall cost. a t t + a 50 t + a 20 t 12 a 2t 35t + 25a = d 35t + 25a 1000 t = 12 and t + a = a = 20 a = 8 (t, a) = (12, 8) This is the Optimum Point d = 35(12) + 25(8) = = 620 The minimum (optimal) cost is $620,000.

8 The school can spend at most 1 million dollars annually on salaries. Maximize the feasible cost. a t t + a 50 t + a 20 t 12 a 2t 35t + 25a = d 35t + 25a 1000 t + a 50 and a = 0 t + 0 = 50 t = 50 (t, a) = (50, 0) This is the Optimum Point d = 35(50) + 25(0) = = 1750 The maximum feasible cost is $1,750,000.

9 Liela Tov bakes cookies for her high school cookie sale. Her chocolate chip cookies sell for $3 a dozen, and her oatmeal brownie cookies sell for $4.50 a dozen. She will bake up to 20 dozen chocolate chip cookies, and up to 40 dozen oatmeal brownie cookies, but no more than 50 dozen cookies, total. Also, the number of oatmeal brownie cookies will be no more than three times the number of chocolate chip cookies. How many of each kind should Liela make in order for the high school to make the most money? How much money will this be?

10 Liela Tov bakes cookies for her high school cookie sale. Her chocolate chip cookies sell for $3 a dozen, and her oatmeal brownie cookies sell for $4.50 a dozen. Let c = a dozen chocolate chip cookies. Let b = a dozen oatmeal brownie cookies. Let P = profit 3c + 4.5b = P

11 She will bake up to 20 dozen chocolate chip cookies, and up to 40 dozen oatmeal brownie cookies, but no more than 50 dozen cookies, total. Also, the number of oatmeal brownie cookies will be no more than three times the number of chocolate chip cookies. Let c = a dozen chocolate chip cookies. Let b = a dozen oatmeal brownie cookies. Let P = profit c 20 b 40 c + b 50 b 3c

12 How many of each kind should Liela make in order for the high school to make the most money? How much money will this be? 55 b Let c = a dozen chocolate chip cookies. Let b = a dozen oatmeal brownie cookies. Let P = profit Liela should make 12.5 dozen chocolate chip cookies and 37.5 dozen oatmeal brownie cookies to make a maximum profit of $ c 3c + 4.5b = P c 20 b 40 c + b 50 b 3c c 3c + b + = 4.5b 50 = b = 50 b 3c = 3c + 4.5b = 120 b = 37.5 c + 3c = 50 3(12.5) + 4.5(37.5) = c = 50 c = 12.5