Question

Objective function Z=3x+6yFind the value at each corner of the graph

Answer 1

Given: An objective function

`z=3x+6y`

with constraints-

`\begin{gathered} \\ \begin{cases}{x\ge0,y\ge0} \\ {2x+y\leq12} \\ {x+y\ge6}\end{cases} \end{gathered}`

Required: To graph the linear inequalities representing the constraints and determine the objective function's value at each corner.

Explanation: The inequalities can be graphed by considering them as equations and then determining the shaded region by less than or greater than symbol.

The equation for the first inequality is-

`2x+y=12`

This represents a straight line passing through points (6,0) and (0,12).

The shaded region will be below this line as the inequality is-

`2x+y\leq12`

Similarly, the inequality-

`x+y\ge6`

Represents a shaded region above the line x+y=6.

The inequalities-

`x\ge0,y\ge0`

Represents the positive values of x and y. Hence we need to determine the graph in the first quadrant.

The graph of the inequalities is-

The graph in blue represents the inequality-

`2x+y\leq12`

While the graph in green represents the inequality-

`x+y\ge6`

The corner points of the common shaded area are A(0,6), B(0,12), and C(6,0).

Now the value of the objective function at these points is-

a) At A(0,6)

`\begin{gathered} z=3(0)+6(6) \\ =36 \end{gathered}`

b) At B(0,12)

`\begin{gathered} z=3(0)+6(12) \\ =72 \end{gathered}`

c) At C(6,0)

`\begin{gathered} z=3(6)+6(0) \\ =18 \end{gathered}`

Final Answer: a) The graph is drawn.

b) 36,72,18