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# Mathematics for Business and Economics MATH 1070

UCONN

GPA 3.97

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This 11 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 1070 at University of Connecticut taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/205829/math-1070-university-of-connecticut in Mathematics (M) at University of Connecticut.

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Date Created: 09/17/15

C hay cer 3 Lintqr Pijmminj Su on 3 W Unur M Obiec ch gym15 M nsm39hs W Rqs ek atan OVH MKZC a Frame w 523 Transhh quasiion m w 0 linear rP mmmiyf wele 32 Sabre lineariuequah cs FQWHC Ryan 31 Linear Programming Problems 152 4 Linear Programming Applications EXAMPLE 1 Maximizing Profits From Benches A shop manufactures two styles of aluminum park benches fancy and plain The fancy style requires 4 kilograms of aluminum and 6 hours of labor while the plain style requires 4 kilograms of aluminum and 3 hours of labor There are 32 kilograms of aluminum and 36 hours of labor available for the day The fancy style bench earns a pro t of 24 and the plain one 18 Naturally the shop s objective is to maximize its pro ts Write a system of inequalities that expresses this information determine the objective function and write all this as a linear programming problem Solution Begin by de ning the variables Let x the number of fancy benches made y the number of plain benches made The information given in the problem is summarized in the table below Fancy Plain Available Kilograms of Aluminum 4 4 32 Hours of Labor 6 3 36 Pro t 24 18 Since each fancy bench requires 4 kilograms of aluminum there is a total of 435 kilograms of aluminum for these benches Since each plain bench requires 4 kilograms of aluminum there are a total of 4y kilograms of aluminum for these benches Then 4x 43 is the total number of kilograms of aluminum needed to make both of the benches while the total amount available is 32 kilograms Thus 4x 4y 5 32 Since each fancy bench requires 6 hours of labor there are a total of 696 hours of labor needed for these benches Since each plain bench requires 3 hours of labor there are a total of 3y hours of labor needed for these benches Then 6m 3y is the total number of hours of labor needed to make both types of the benches while the total labor available is 36 hours Thus 6m 3y 5 36 39 Naturally we must have a 2 0 and y 2 0 which we refer to as the nonnegativity constraint We then have the system of inequalities con straints 4m 4y 3 32 kilograms of aluminum constraint 6x 3y S 36 hours of labor constraint 903 2 0 nonnegativity constraint The pro t function in dollars is P 2400 18y Our objective is to maximize this function Maximize P 2405 18y objective function Subject to 4m 43 S 32 kilograms of aluminum 65 33 S 36 hours of labor cc y 2 0 nonnegativity Notice that we now have several possibilities for the number of benches produced For example the shop could make one fancy bench and one plain X 393 Emmy W X2 a cauan L X2 23 CWKL X 3 chj V Q VA Fragsquot 2 8 1 421 43 32 6X 33 36 P 24x x33 xa 537o x 2 4 X 4 0 x r 4r D L s Avai 2 L O quotitsng 4quot 9 SO fq m c 2 gt a So i 6b W 980 4 X AO OP D5 7 Vw oFs 1x43580 X7t 4x23 650 370 2x 23 5 so 1939 eox rso l 3x13 4 010 p X 3xch Zje 3X 3M 7 5 v 3 3 2x Yrquot X Z Bxk j l8 53 77 AX f39bgc C anaba c ax Haa C mH laa 7 177 c Xj 4 374 M55 i 3 4 z a A Hjsz 39 4x43582 470 6x 34 3G 8 7 4xqj32 6X3J 3G 43 4X 3932 3yc 6X RZQ j quotY 5 quot x 112 4 xwiwn mXV I gt35 a F m 15 T 439 5 ammva A D 352 H n 0 3 m xM V im m3 Swaia quot4 Ll l n n 5 I l

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