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Question Number 50519 by Tawa1 last updated on 17/Dec/18

Answered by afachri last updated on 17/Dec/18

Commented by afachri last updated on 17/Dec/18

the satisfied region of inequation is the white one. Blue line ⇒ x+y ≤ 24 Red line ⇒ 50x+40y ≤ 1050 Purple line ⇒ x ≥ 0 Black line ⇒ y ≥ 0

$$\mathrm{the}\:\mathrm{satisfied}\:\mathrm{region}\:\mathrm{of}\:\mathrm{inequation}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{white}\:\mathrm{one}. \\ $$$$\mathrm{Blue}\:\mathrm{line}\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:{x}+{y}\:\:\:\leqslant\:\mathrm{24} \\ $$$$\mathrm{Red}\:\mathrm{line}\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\mathrm{50}{x}+\mathrm{40}{y}\:\:\leqslant\:\mathrm{1050} \\ $$$$\mathrm{Purple}\:\mathrm{line}\:\:\:\Rightarrow\:\:\:\:{x}\:\geqslant\:\mathrm{0} \\ $$$$\mathrm{Black}\:\mathrm{line}\:\:\:\:\:\:\Rightarrow\:\:\:{y}\:\geqslant\:\mathrm{0} \\ $$

Answered by afachri last updated on 17/Dec/18

let : a bottle of cola : x a bottle of lemonade : y 50x + 40y ≤ 1050 x + y ≤ 24 x ≥ 0 ; y≥0

$$\boldsymbol{\mathrm{let}}\:: \\ $$$$\:\mathrm{a}\:\mathrm{bottle}\:\mathrm{of}\:\mathrm{cola}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\:{x} \\ $$$$\:\mathrm{a}\:\mathrm{bottle}\:\mathrm{of}\:\mathrm{lemonade}\:\:\:\:\:\::\:{y} \\ $$$$ \\ $$$$\:\mathrm{50}{x}\:+\:\mathrm{40}{y}\:\:\leqslant\:\mathrm{1050} \\ $$$$\:\:\:\:\:\:{x}\:\:+\:\:{y}\:\:\:\:\:\:\:\leqslant\:\:\:\:\:\:\:\mathrm{24} \\ $$$$\:{x}\:\geqslant\:\mathrm{0}\:\:\:;\:\:\:{y}\geqslant\mathrm{0} \\ $$$$ \\ $$

Answered by afachri last updated on 17/Dec/18

Profit Function Z = 10x + 5y (0 , 24) ⇒ Z = 10(0) + 5(24) = N120 (9 , 15) ⇒ Z = 10(9) + 5(15) = 90 + 75 = N165 (21 , 0) ⇒ Z = 10(21) + 5(0) = N210 conclusion : the woman should buy 21 bottles of Cola to earn maximum profit.

$$\mathrm{Profit}\:\mathrm{Function} \\ $$$$\boldsymbol{\mathrm{Z}}\:=\:\mathrm{10}{x}\:+\:\mathrm{5}{y} \\ $$$$\left(\mathrm{0}\:,\:\mathrm{24}\right)\:\:\:\Rightarrow\:\:\:\:\boldsymbol{\mathrm{Z}}\:=\:\mathrm{10}\left(\mathrm{0}\right)\:+\:\mathrm{5}\left(\mathrm{24}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\:\:\mathrm{N120} \\ $$$$\left(\mathrm{9}\:,\:\mathrm{15}\right)\:\:\:\Rightarrow\:\:\:\:\boldsymbol{\mathrm{Z}}\:=\:\mathrm{10}\left(\mathrm{9}\right)\:+\:\mathrm{5}\left(\mathrm{15}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{90}\:+\:\mathrm{75} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{N165}\: \\ $$$$\left(\mathrm{21}\:,\:\mathrm{0}\right)\:\:\:\Rightarrow\:\:\:\:\boldsymbol{\mathrm{Z}}\:=\:\mathrm{10}\left(\mathrm{21}\right)\:+\:\mathrm{5}\left(\mathrm{0}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{N210} \\ $$$$\boldsymbol{\mathrm{conclusion}}\::\:\mathrm{the}\:\mathrm{woman}\:\mathrm{should}\:\mathrm{buy}\:\mathrm{21}\:\mathrm{bottles} \\ $$$$\mathrm{of}\:\mathrm{Cola}\:\mathrm{to}\:\mathrm{earn}\:\mathrm{maximum}\:\mathrm{profit}. \\ $$

Commented by Tawa1 last updated on 19/Dec/18

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by afachri last updated on 19/Dec/18

Ur welcome..

$$\mathrm{Ur}\:\mathrm{welcome}.. \\ $$

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