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The factory will create a lot of noise and fumes. The area is one with a high level of unemployment - people's incomes are below the national average. Before deciding whether or not to grant planning permission the government also considers the external costs and external benefits - that is, the impact on the rest of society other than the business itself.

External costsExternal benefitsWaste products will cause pollutionJobs will be createdSmoke and fumes may damage the health of residentsOther firms may move into the area to provide services to the chemical firmParkland cannot now be used by local residentsThe chemical factory will pay taxes - government might increase spending on social projects e.g. more hospitals\begin{matrix} \text{External costs} & \text{External benefits}\\ \text{Waste products will cause pollution} & \text{Jobs will be created}\\ \text{Smoke and fumes may damage the health of residents} & \text{Other firms may move into the area to provide services to the chemical firm}\\ \text{Parkland cannot now be used by local residents} & \text{The chemical factory will pay taxes - government might increase spending on social projects e.g. more hospitals}\\ \end{matrix}

The government will try to give a value to all of these costs and benefits. This is called cost-benefit analysis. This is not easy to do. For example, what is the cost of losing parkland for children to play on? Estimates are made and then the total costs and benefits of the decision are added up: private costs, added to external costs, give a total social cost, private benefits, added to external benefits, give a total social benefit figure. If the total social benefit is greater than the total social cost, the scheme is likely to be accepted. If, however, the total social cost is greater than the total social benefit, the government will probably refuse permission. Explain why any two stakeholder groups will be worried about the external costs from a new chemical factory.


Find a value of a so that each objective function has maximum values at the indicated vertex, subject to the following constraints.

x0y2x+y94x+3y6\begin{aligned} &\begin{array}{l} {x \geq 0} \\ {y \geq 2} \end{array}\\ &x+y \leq 9\\ &-4 x+3 y \leq 6 \end{aligned}

f(x,y)=4x+ay,(3,6)f(x, y)=-4 x+a y,(3,6)


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Plot the Inequalities and Identify the vertices of common region :

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