$3.$ Each coffee table produced by Kevin Watson Designers nets the firm a profit of $\backslash (\backslash$$ $9\backslash ).$ Each bookcase yields a $\backslash (\backslash$$ $15\backslash )$ profit. Watson's firm is small and its resources limited$.$ During any given production period $($of $1$ week$),8$ gallons of varnish and $14$ lengths of high$-$quality redwood are available. Each coffee table requires approximately $1$ gallon of varnish and $1$ length of redwood. Each bookcase takes $1$ gallon of varnish and $2$ lengths of wood.

The aim of the objective function for Kevin should be to the objective value.

Decision variables:

$\backslash ($ X$=\backslash )$ number of coffee tables produced $/$ week

$\backslash (\backslash$mathrm$\{$Y$\}=\backslash )$ number of bookcases produced $/$ week

Objective function:

$\backslash [$

$\backslash$mathrm$\{$Z$\}=$

$\backslash ]$

Subject to:

$\backslash [$

$\backslash$begin$\{$aligned$\}$

$1$ X$+1$ Y & $\backslash$leq $8\backslash \backslash$

$1$ X$+2$ Y & $\backslash$leq $14\backslash \backslash$

X$,$ Y $\backslash$geq $0$ &

$\backslash$end$\{$aligned$\}$

$\backslash ]$

On the graph on right, constraints $\backslash (\backslash$mathrm$\{$C$\}\_\{1\}\backslash )$ and $\backslash (\backslash$mathrm$\{$C$\}\_\{2\}\backslash )$ have been plotted.

Using the point drawing tool, plot all the corner points for the feasible area.

The optimum solution is:

$\backslash ($ X$=\backslash )($round your response to two decimal places$).$

$\backslash (\backslash$mathrm$\{$Y$\}=\backslash )($round your response to two decimal places$).$

Optimal solution value $\backslash ($ Z$=\backslash )($round your response to two decimal places$).$