A craftsman named William Barnes builds two kinds of

birdhouses, one for wrens and a second for bluebirds. Each wren birdhouse takes $4$ hours of

labor and $4$ units of lumber. Each bluebird house requires $2$ hours of labor and $12$ units of

lumber. The craftsman has available $48$ hours of labor and $120$ units of lumber. Wren houses

yield a profit of $$10$ each and bluebird houses yield a profit of $$15$ each.

The aim of the objective function for William should be to

the objective value.

Decision variables:

$x=$ number of wren houses to be produced

$Y=$ number of bluebird houses to be produced

a$)$ Objective function:

$Z=x+Y$

Subject to:

$\{\begin{array}{ccc}\text{:}[4x+2Y,& \text{g}\text{r}\text{a}\text{d}48,& (C_1)]\end{array}$

$\{\begin{array}{ccc}\text{:}[4x+12Y,& \text{g}\text{r}\text{a}\text{d}\text{,}120,& (C_2)]\end{array}$

$x,Y0$

b$)$ On the graph on right, constraints $C_1$ and $C_2$ have been plotted.

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

The optimum solution is:

$x=(round$ your response $to$ two decimal places$).$

$Y=(round$ your response $to$ two decimal places$).$

Optimal solution value $\text{'}$Z$\text{'}=,($round your response to two decimal places$).$