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Tentukan nilai minimum C = 30 x + 10 y dengan syarat: 1 , 8 x + 0 , 9 y ≥ 270 ; 0 , 3 x + 0 , 2 y ≥ 54 ; 0 , 01 x + 0 , 03 y ≥ 3 , 9 ; x ≥ 0 ; y ≥ 0 .

Tentukan nilai minimum  dengan syarat: .

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I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

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Pembahasan

Penentuan titik potong masing-masing garis pembatas dengan sumbu koordinat, seperti terlihat pada tabel di bawah ini. Penentuan titik ekstrim Misal titik A adalah perpotongan garis dan garis , maka koordinat titik A adalah Misal titik Badalah perpotongan garis dan garis , maka koordinat titik B adalah Misal titik Cadalah perpotongan garis dan garis maka koordinat titik Cadalah Penentuan nilai minimum dengan uji titik pojok Jadi, nilai minimum adalah 3600

Penentuan titik potong masing-masing garis pembatas dengan sumbu koordinat, seperti terlihat pada tabel di bawah ini.

Penentuan titik ekstrim

Misal titik A adalah perpotongan garis 1 comma 8 x plus 0 comma 9 y equals 270 dan garis 0 comma 3 x plus 0 comma 2 y equals 54, maka koordinat titik A adalah

table row cell 1 comma 8 x end cell plus cell 0 comma 9 y end cell equals 270 row cell 0 comma 3 x end cell plus cell 0 comma 2 y end cell equals 54 end table open vertical bar table row cell cross times 1 end cell row cell cross times 6 end cell end table close vertical bar table row blank blank blank blank blank blank row cell 1 comma 8 x end cell plus cell 0 comma 9 y end cell equals 270 blank row cell 1 comma 8 x end cell plus cell 1 comma 2 y end cell equals 324 minus row blank blank cell negative 0 comma 3 y end cell equals cell negative 54 end cell blank end table space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y space space equals space 180

table attributes columnalign right center left columnspacing 0px end attributes row cell 0 comma 3 x plus 0 comma 2 open parentheses 180 close parentheses end cell equals 54 row cell 0 comma 3 x end cell equals cell 54 minus 36 end cell row cell 0 comma 3 x end cell equals 18 row x equals 60 end table

Misal titik B adalah perpotongan garis 1 comma 8 x plus 0 comma 9 y equals 270 dan garis size 14px 0 size 14px comma size 14px 01 size 14px x size 14px plus size 14px 0 size 14px comma size 14px 03 size 14px y size 14px greater or equal than size 14px 3 size 14px comma size 14px 9, maka koordinat titik B adalah

table row cell 1 comma 8 x end cell plus cell 0 comma 9 y end cell equals 270 row cell 0 comma 01 x end cell plus cell 0 comma 03 y end cell equals cell 3 comma 9 end cell end table open vertical bar table row cell cross times 1 end cell row cell cross times 180 end cell end table close vertical bar table row blank blank blank blank blank blank row cell 1 comma 8 x end cell plus cell 0 comma 9 y end cell equals 270 blank row cell 1 comma 8 x end cell plus cell 5 comma 4 y end cell equals 702 minus row blank blank cell 4 comma 5 y end cell equals 432 blank end table space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y space space equals space 96

table attributes columnalign right center left columnspacing 0px end attributes row cell 0 comma 01 x plus 0 comma 03 open parentheses 96 close parentheses end cell equals cell 3 comma 9 end cell row cell 0 comma 01 x end cell equals cell 3 comma 9 minus 2 comma 88 end cell row x equals cell fraction numerator 1 comma 02 over denominator 0 comma 01 end fraction end cell row blank equals 102 end table

Misal titik C adalah perpotongan garis 0 comma 3 x plus 0 comma 2 y equals 54dan garis size 14px 0 size 14px comma size 14px 01 size 14px x size 14px plus size 14px 0 size 14px comma size 14px 03 size 14px y size 14px greater or equal than size 14px 3 size 14px comma size 14px 9maka koordinat titik C adalah

table row cell 0 comma 3 x end cell plus cell 0 comma 2 y end cell equals 54 row cell 0 comma 01 x end cell plus cell 0 comma 03 y end cell equals cell 3 comma 9 end cell end table open vertical bar table row cell cross times 1 end cell row cell cross times 30 end cell end table close vertical bar table row blank blank blank blank blank blank row cell 0 comma 3 x end cell plus cell 0 comma 2 y end cell equals 54 blank row cell 0 comma 3 x end cell plus cell 0 comma 9 y end cell equals 117 minus row blank blank cell negative 0 comma 7 y end cell equals cell negative 63 end cell blank end table space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y space equals space 90

table attributes columnalign right center left columnspacing 0px end attributes row cell 0 comma 3 x plus 0 comma 2 open parentheses 90 close parentheses end cell equals 54 row cell 0 comma 3 x end cell equals cell 54 minus 18 end cell row x equals cell fraction numerator 36 over denominator 0 comma 3 end fraction end cell row blank equals 120 end table

Penentuan nilai minimum dengan uji titik pojok

table attributes columnalign right center left columnspacing 0px end attributes row straight C equals cell 30 x plus 10 y end cell row cell straight A open parentheses 60 comma 180 close parentheses end cell rightwards double arrow cell straight C equals 30 open parentheses 60 close parentheses plus 10 open parentheses 180 close parentheses equals 3600 end cell row cell straight B open parentheses 102 comma 96 close parentheses end cell rightwards double arrow cell straight C equals 30 open parentheses 102 close parentheses plus 10 open parentheses 96 close parentheses equals 4020 end cell row cell straight C open parentheses 120 comma 90 close parentheses end cell rightwards double arrow cell straight C equals 30 open parentheses 120 close parentheses plus 10 open parentheses 90 close parentheses equals 4500 end cell end table

Jadi, nilai minimum begin mathsize 14px style straight C equals 30 x plus 10 y end style adalah 3600

 

 

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