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 60 hours of labor and 120 units of lumber. Wren houses yield a profit of $6 each, and bluebird houses yield a profit of $15 each. a) Write out the objective and constraints. b) Solve graphically.

a) 6W+15B

subject to:

4W+2B ≤ 60

4W + 12B≤ 120

W>0

B>0

b) see attachment

Explanation:

a) W: wren birdhouse

B: bluebird house

Objective function:

6W+15B

Explicit constraints:

4W+2B ≤ 60

4W + 12B≤ 120

Implicit constraints

W>0

B>0

b)  coordinates of optimal region:

(0,0), (0,10), (12,6), (15,0)

For optimum profit:

(0,0): 6(0) + 15(0)= 0

(0,10): 6(0) + 15(10)= 150

(12,6): 6(12) + 15(6)= 162

(15,0): 6(15) + 15(0)= 90

Optimal solution is: (12,6) or 12 wren birdhouse and 6 bluebird house

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