numerical methods finals

7/28/2019 numerical methods finals

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ES 84LAB 3

(Real-world problem/application which can be solved using

numerical methods)

CALAPIZ, Jebien MayDAJAO, Demie MaeGARCIA, Christina

LABADAN, MenchieLADERA, Gerewin Ed

LONGAKIT, Pleides FaithTABAAG, Darlyn


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APPLICATION OF NUMERICAL METHODS(BUSINESS)

CASE STUDY PROBLEM

Way back in the year 80s, a merchant in Butuan has three itemson sale, namely a radio for Php50.00, a clock for Php30.00, and a

flashlight for Php1.00. At the end of his usual days, he often solds a totalof 100 of the three items and takes exactly Php1000.00 from his sales.The merchant then thought and wished to increase his earning byincreasing his supplies. He has noticed a 4:1:20 percentage ratio betweenhis items, i.e. the radio, clock and flashlight respectively, on his records, inwhich he too found out would give the same digit (4) if the mentioned

percentage ratio is multiplied to its respective item. If the merchant alsodiscovered that the constant digit is directly proportional to his earningfactor. i.e. twice his earning makes the number 8, how many of eachproduct does he need to purchase for his stocks to double his earnings?How many of each item did he usually able to put on the market?


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Five months later, a new store nearby had opened and had soldthe same in demand products causing a threat to the merchants industry.If the competitor takes away half of the merchants sales, then how many

o f e a c h o f t h e m e r c h a n t s p r o d u c t s w a s t h e n s o l d ?The total time it takes the merchant to sell his products isdetermined by the function F(T). Find the selling rate of each item giventhe following parameters:

where

Tx= the selling rate of radio

Ty=the selling rate of clock

Tz=the selling rate of flashlight

TIME(hrs)

Function, F(T)

T(1/2) (for half earnings) F(T(1/2))= 1.83334

T (for regular sales) F()= 2.57833

T(2) (for double earnings) F(T(2))= 11.975

F(T(1/2))= (1.5x) Tx + (y^2) Ty (z /10) TzF()= (x/16) Tx + (y+1) Ty + (o.0333z) TzF(T(2))=(0.10x) Tx + (y) Ty + (z/12) Tz


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III. OBJECTIVE

To solve the real-world problem on businesssuch as the query on the number of items sold and the sellingrates with respect to the entrepreneurs earnings using

numerical methods specifically that of the simultaneous linearequations, LU and Gauss Seidel.

IV. HYPOTHESIS

Given the parameters and conditions of themerchants production. we say that the earnings are directly

proportional to the number of items to be sold. As such, thenumber of items being sold determines the selling rate of eachproduct, respectively.


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Let x be the number of radioLet y be the number of clockLet Z be the number of flashlight

@ usual sales/earnings , Tx + y + z =10050x + 30y + z =1000 x + (1)y + 1/20 z = 3(4) -- 0.25x + y + 0.05z =12

V. METHODOLOGY

Analysis :First, solve for the number of each item sold at normal sales, doubleearnings and at half of his earnings when a competitor arrived 5 months after.Then we have :


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@ double earnings , T(2)

x + y + z =100(2) -- >x + y +z = 20050x + 30y + z =1000(2) -- > 50x + 30y + z = 2000 x + (1)y + 1/20 z = 3(8)-- 0.25x + y + 0.05z =24

@ half earnings, T(1/2)

x + y + z =100(1/2) -- >x + y +z = 5050x + 30y + z =1000(1/2) -- > 50x + 30y + z = 500 x + (1)y + 1/20 z = 3(2) 4) -- 0.25x + y + 0.05z =6


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>solving the respective x, y,z at each earning (using LU), wehave the results:

@ usual sales/earnings , Tx= 16 , y=4 , z=80

@ double earnings , T(2)x= 32 , 16 , y=8 , z=160

@ half earnings, T(1/2)x= 8 , 16 , y=4 , 4 , z=40

Then after which, proceed to determining the selling rateTx, Ty, and Tz using the given parameters through thesimultaneous linear equations,Gauss Seidel.


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Given the following functions for each earnings:F(T(1/2))= (1.5x) Tx + (y^2) Ty (z /10) TzF()= (x/16) Tx + (y+1) Ty + (o.0333z) TzF(T(2))=(0.10x) Tx + (y) Ty + (z/12) Tz

We substitute the solved values for x,y, and z for the respective cases ofearnings, we then have:

@ half earnings, T(1/2)F(T(1/2))=12Tx + 4Ty 4Tz1.83334=12Tx + 4Ty 4Tz

@ usual sales/earnings , TF()= Tx + 5Ty + 2.6664 Tz2.57833= Tx + 5Ty + 2.6664 Tz

@ double earnings , T(2)F(T(2))= 3.2 Tx + 8Ty + 13.33 Tz11.975= 3.2 Tx + 8Ty + 13.33 Tz

Now, we have another set of linear equations for the selling rates of theradio,clock and flashlight respectively. We solve Tx, T y, and Tz usingGauss seidel.


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VI. RESULTS AND DISCUSSIONSSolving the respective x, y,z at each earning (using LU), we have the

results:

@ usual sales/earnings , T


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@ double earnings , T(2)


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@ half earnings, T(1/2)


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Determining the selling rate Tx, Ty, and Tz using the given parametersthrough the simultaneous linear equations,Gauss Seidel, we have :


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SUMMARY OF RESULTS

@ usual sales/earnings , Tx= 16 radios, y=4 clocks and z=80 flashlights

@ double earnings , T(2)x= 32radios, y= 16 clocks, z=160 flashlights

@ half earnings, T(1/2)x= 8 radios , y= 2 clocks, z= 40 flashlights

Determining the selling rate Tx, Ty, and Tz using the given parametersthrough the simultaneous linear equations,Gauss Seidel, we have :

Solving the respective x, y,z at each earning (using LU), we have the results:

Tx= 0.4137756, Ty=0.0100158, Tz=0.7930076

Err.x= 0.0000020, Err. y= 0.0000036, Err. z= 0.0000003


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TCCWC

XOXO

Greetings from Mr. Bench

numerical methods finals

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