Differensial 3 barang

DIFFERENSIAL 3 BARANG

SOAL :

Diketahui        :

C = 0,050Qa² + Qb² + 0,75Qc² + 2QaQb + 3QbQc + QaQc +9187,50

Persamaan fungsi permintaan :

Pa   = 325 – 4Qa

Pb   = 560 – 5Qb

Pc   = 355 – 3Qc

Diminta :

a.       Jumlah factor produksi masing-masing barang

b.      Harga Per unit masing-masing barang

c.       Besarnya laba maksimum tersebut

Jawab :

Cara I :

R  = P(Q)

Ra= Pa(Qa)

Ra= 325 – 4Qa (Qa)  ®   -4Qa² + 325Qa

Rb= 560 – 5Qb (Qb)   ®   -5Qb² + 560Qb

Rc= 355 – 3Qc (Qc)   ®   -3Qc² + 355Qc

    = -4Qa² + 325Qa - 5Qb² + 560Qb – 3Qc² + 355Qc – (0,050Qa² + Qb² + 0,75Qc² + 2QaQb + 3QbQc + QaQc +9187,50)

    = -4Qa² + 325Qa - 5Qb² + 560Qb – 3Qc² + 355Qc – 0,050Qa² - Qb² - 0,75Qc² - 2QaQb - 3QbQc - QaQc - 9187,50

    = -4,5Qa² + 325Qa - 6Qb² + 560Qb – 3,75Qc² + 355Qc - 2QaQb - 3QaQc - QaQc - 9187,50

Jadi

p a = -4,5Qa² + 325Qa - 2QaQb – QaQc

pa¹ = -9Qa + 325 – 2Qb – Qc                         ®        -9Qa – 2Qb – Qc = -325       ( Persamaan I)

pb  = - 6Qb² + 560Qb - 2QaQb - 3QbQc

pb¹= -12Qb + 560 – 2Qa – 3Qc                     ®        -2Qa – 12Qb – 3Qc = -560   (Persamaan II)

pc = -3,75Qc² + 355Qc - 3QbQc - QaQc

pc¹= -7,5Qc + 355 – 3Qb – Qa                       ®        – Qa  – 3Qb - 7,5Qc = -355  (Persamaan III)

Eliminasi Persamaan I & II :

-9Qa – 2Qb – Qc = -325           x 6 ® -54Qa² – 12Qb- 6Qc = -1.950

2Qa – 12Qb – 3Qc = -560        x 1 ® -2Qa² -12Qb – 3Qc  = -560

                                                             -52Qa² -3Qc             = -1.390                                  (Persamaan IV)

   2Qa – 12Qb – 3Qc = -560     x 1 ® -2Qa² – 12Qb- 3Qc    = -560

 – Qa  – 3Qb - 7,5Qc = -355     x 4 ® -4Qa² - 12Qb – 30Qc = -1.420

                                                               2Qa² - 27Qc             = 860                         (Persamaan V)

Eliminasi persamaan IV & V :

-52Qa - 3Qc   = -1.390             x  9 ® -468Qa – 27Qb - 3Qc  = -12.510

   2Qa - 27Qc = 860                  x 1 ®      2Qa + 27Qb            =  860

                                                                                    -466Qa  = -11.650      

                                                                                           Qa  = 25

2Qa² - 27Qc  = 860                                         – Qa  – 3Qb - 7,5Qc = -355   

            27Qc = 860 – 50                                                         -3Qb = -355 + 25 + 225

                Qc = 30                                                                       Qb = 35

a)      Jumlah produksi

Qa = 25

Qb = 35

Qc = 30

b)      Harga jual atau unit

Pa = 325 – 4(Qa)                          Pb = 560 – 5(Qb)                                Pc = 355 – 3(Qc)

Pa = 325 – 4(25)                          Pb = 560 – 5(35)                                 Pc = 355 – 3(30)

Pa =325 – 100                              Pb = 560 - 175                                                Pc = 355 - 90

Pa = 225                                       Pb = 385                                              Pc = 265

c)      Laba

Qa = 25

Qb = 35

Qc = 30

p = -4,5Qa² + 325Qa - 6Qb² + 560Qb – 3,75Qc² + 355Qc - 2QaQb - 3QaQc - QaQc – 9187,50

p = -2.812,5 = 8.125 – 7.350 + 19.600 + 3.375 +  10.650 -1.750 – 3.150 – 750 – 9.187,50

p = 10.000

Cara II :

Ra= -4Qa² + 325Qa

Rb= -5Qb² + 560Qb

Rc=  -3Qc² + 355Qc

Ra^’  = Mra

Mra = -8Qa + 325

Mrb = -10Qb + 560

Mrc = -6Qc + 355

C = 0,050Qa² + Qb² + 0,75Qc² + 2QaQb + 3QbQc + QaQc +9187,50

C’= Mc

Mca= Qa + 2qb +Qc               ® Mca = Qa + 2Qb + Qc

Mcb= 2Qb + 2Qa + 3Qc         ® Mcb = 2Qa + 2Qb + 3Qc

Mcc= 1,5Qc + 3Qb + Qa        ® Mcc  = Qa + 3Qb + 1,5Qc

pmaks a ®      Mra = Mca

-8Qa + 325 = Qa + 2Qb + Qc

            325 = Qa + 8Qa + 2Qb + Qc 

                        325 = 9Qa + 2Qb + Qc                                               (Persamaan I)

pmaks b ®      Mrb  = Mcb

-10Qb + 560 = 2Qa + 2Qb + 3Qc

  560 = 2Qa + 2Qb + 10Qb + 3Qc

                          560 = 2Qa + 12Qb + 3Qc                                         (Persamaan II)

pmaks c ®      Mrc = Mcc

-6Qc + 355 = Qa + 3Qb + 1,5Qc

355 = Qa + 3Qb + 1,5Qc + 6Qc

                        355 = Qa + 3Qb + 7,5Qc                                            (Persamaan III)

Eliminasi pers I&II

325 = 9Qa + 2Qb + Qc           x3  ®  975 = 27Qa + 6Qb + 3Qc

560= 2Qa + 12Qb + 3Qc        x1  ®  560 = 2Qa + 12Qb + 3Qc

                                                            415 = 25Qa – 6Qb                  (Persamaan IV)

Eliminasi pers II&III

560 = 2Qa + 12Qb + 3Qc       x5  ®  2800 = 10Qa+60Qb+15Qc

355 = Qa + 3Qb + 7,5Qc        x2  ®    710 = 2Qa+6Qb+15Qc

                                                            2090 = 8Qa=54Qb                  (Persamaan V)

Eliminasi pers IV& V

415 = 25Qa – 6Qb                  x9  ®  3735 = 225Qa - 54Qb

2090 = 8Qa=54Qb                  x1  ®    710 = 2Qa+6Qb+15Qc

                                                            5825 = 233Qa

                                                                25 = Qa

415 = 25Qa – 6Qb                              560 = 2Qa + 12Qb + 3Qc      

415 = 25 (25) – 6Qb                           560 = 2(25) + 12(35) + 3Qc

 415 – 625  = -6Qb                              560-50-420 = 3Qc

            35   = Qb                                               30  = Qc

a)      Jumlah produksi

Qa = 25

Qb = 35

Qc = 30

b)      Harga jual atau unit

Pa = 325 – 4(Qa)                          Pb = 560 – 5(Qb)                                Pc = 355 – 3(Qc)

Pa = 325 – 4(25)                          Pb = 560 – 5(35)                                 Pc = 355 – 3(30)

Pa =325 – 100                              Pb = 560 - 175                                                Pc = 355 - 90

Pa = 225                                       Pb = 385                                              Pc = 265

c)      Laba

Qa = 25

Qb = 35

Qc = 30

p = -4,5Qa² + 325Qa - 6Qb² + 560Qb – 3,75Qc² + 355Qc - 2QaQb - 3QaQc - QaQc – 9187,50

p = -2.812,5 = 8.125 – 7.350 + 19.600 + 3.375 +  10.650 -1.750 – 3.150 – 750 – 9.187,50

p = 10.000