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Question Number 32242 by pieroo last updated on 22/Mar/18
A company manufactures two types of products;  X ($4.50 profit per item x) and Y ($3.00 profit per  item y). These items are built using both machine  time and manual labour. The X product requires  3 hours of machine time and two hours of manual  labour. The Y product requires 3 hours of machine  time and no manual labour. If the week′s supply of   manual labour is limited to 8 hours and machine  time to 15 hours, write down all inequalities   involving x and y.
$$\mathrm{A}\:\mathrm{company}\:\mathrm{manufactures}\:\mathrm{two}\:\mathrm{types}\:\mathrm{of}\:\mathrm{products}; \\ $$$$\mathrm{X}\:\left(\$\mathrm{4}.\mathrm{50}\:\mathrm{profit}\:\mathrm{per}\:\mathrm{item}\:\mathrm{x}\right)\:\mathrm{and}\:\mathrm{Y}\:\left(\$\mathrm{3}.\mathrm{00}\:\mathrm{profit}\:\mathrm{per}\right. \\ $$$$\left.\mathrm{item}\:\mathrm{y}\right).\:\mathrm{These}\:\mathrm{items}\:\mathrm{are}\:\mathrm{built}\:\mathrm{using}\:\mathrm{both}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{and}\:\mathrm{manual}\:\mathrm{labour}.\:\mathrm{The}\:\mathrm{X}\:\mathrm{product}\:\mathrm{requires} \\ $$$$\mathrm{3}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{machine}\:\mathrm{time}\:\mathrm{and}\:\mathrm{two}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{manual} \\ $$$$\mathrm{labour}.\:\mathrm{The}\:\mathrm{Y}\:\mathrm{product}\:\mathrm{requires}\:\mathrm{3}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{and}\:\mathrm{no}\:\mathrm{manual}\:\mathrm{labour}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{week}'\mathrm{s}\:\mathrm{supply}\:\mathrm{of}\: \\ $$$$\mathrm{manual}\:\mathrm{labour}\:\mathrm{is}\:\mathrm{limited}\:\mathrm{to}\:\mathrm{8}\:\mathrm{hours}\:\mathrm{and}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{to}\:\mathrm{15}\:\mathrm{hours},\:\mathrm{write}\:\mathrm{down}\:\mathrm{all}\:\mathrm{inequalities}\: \\ $$$$\mathrm{involving}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}. \\ $$
Answered by Joel578 last updated on 22/Mar/18
 { ((3x + 3y ≤ 15)),((2x ≤ 8)),((x ≥ 0)),((y ≥ 0)) :}
$$\begin{cases}{\mathrm{3}{x}\:+\:\mathrm{3}{y}\:\leqslant\:\mathrm{15}}\\{\mathrm{2}{x}\:\leqslant\:\mathrm{8}}\\{{x}\:\geqslant\:\mathrm{0}}\\{{y}\:\geqslant\:\mathrm{0}}\end{cases} \\ $$
Commented by pieroo last updated on 22/Mar/18
thanks very much
$$\mathrm{thanks}\:\mathrm{very}\:\mathrm{much} \\ $$

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