### Linear Programming

Question Sample Titled 'Linear Programming'

 (a) In the figure, the shaded region (including the boundary) is bounded by straight lines ${L}_{{1}}$ , ${L}_{{2}}$ , the ${x}$-axis and the ${y}$-axis. ${L}_{{1}}$ and ${L}_{{2}}$ intersect at ${\left({4},{4}\right)}$ . Find the system of inequalities representing the shaded region. (4 marks)

${x}$${y}$${\left({4},{4}\right)}$${6}$${10}$${L}_{{1}}$${L}_{{2}}$${O}$

 (b) Bianca wants to produce two types of synthetic paper: synthetic paper ${A}$ and synthetic paper ${B}$ . The sum of the total weight of synthetic paper ${B}$ and ${2}$ times of the total weight of synthetic paper ${A}$ cannot be more than ${12}$ $\text{kg}$ . The cost of producing synthetic paper ${A}$ is $40$40$$/$$\text{kg}$ and the cost of producing synthetic paper ${B}$ is$60$60$$/$$\text{kg}$ . The total cost of synthetic paper ${A}$ and synthetic paper ${B}$ cannot be more than $400$400$ . Suppose that the profits of selling ${1}$ $\text{kg}$ of synthetic paper ${A}$ and synthetic paper ${B}$ are$85$85$  and $125$125$  respectively. Bianca claims that the total profit$Z$Z$  can be more than $860$860$  . Do you agree? Explain your answer. (4 marks) 題解  (a) Equation of ${L}_{{1}}:$ $\dfrac{{{4}-{0}}}{{{4}-{6}}}$ $=\dfrac{{{y}-{0}}}{{{x}-{6}}}$ 1M ${2}{x}+{y}-{12}$ $={0}$ 1A Equation of ${L}_{{2}}:$ $\dfrac{{{4}-{0}}}{{{4}-{10}}}$ $=\dfrac{{{y}-{0}}}{{{x}-{10}}}$ ${2}{x}+{3}{y}-{20}$ $={0}$ 1A The system of inequalities: ${2}{x}+{y}-{12}$ $\le{0}$ 1A ${2}{x}+{3}{y}-{20}$ $\le{0}$ ${x}$ $\ge{0}$ ${y}$ $\ge{0}$ (b) Let ${x}$ $\text{kg}$ and ${y}$ $\text{kg}$ be the weights of synthetic paper ${A}$ and synthetic paper ${B}$ produced respectively. From the given information, we have ${2}{x}+{y}$ $\le{12}$ 1A ${40}{x}+{60}{y}$ $\le{400}$ ${x}$ $\ge{0}$ ${y}$ $\ge{0}$ ${2}{x}+{y}-{12}$ $\le{0}$ ${2}{x}+{3}{y}-{20}$ $\le{0}$ ${x}$ $\ge{0}$ ${y}$ $\ge{0}$ The objective function is ${Z}$ $={85}{x}+{125}{y}$ 1A The ${y}$-intercept of ${L}_{{2}}$ is $\dfrac{{20}}{{3}}$ . Consider the vertices of shaded region. At ${\left({0},{0}\right)},{Z}={0}$ 1M At ${\left({6},{0}\right)},{Z}={510}$ At ${\left({0},\dfrac{{20}}{{3}}\right)},{Z}=\dfrac{{2500}}{{3}}$ At ${\left({4},{4}\right)},{Z}={840}$ Thus, the maximum profits is$840$840$  . Therefore, Bianca is disagreed. 1A

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