Question

A woodworking company handcrafts two types of birdhouses. One is large and the other is small. Both types of houses go through two technicians; first a shaper and then a painter. The large house requires two hours of the shaper's time and one hour of the painter's time. The small house requires one hour of the shaper's time and two hours of the painter's time. The shaper has 104 hours and the painter has 76 hours of time available this month. Profit on the large house is $6 and on the small house is $11. The following inequalities are formed from the information provided: Let x = large bird houses and y = small bird houses. x ≥ 0, y ≥ 0 2x + y ≤ 104 (shaper) x + 2y ≤ 76 (painter) Assuming that all the houses that are produced can be sold, how many of each type of house should the company manufacture per month to obtain the best returns?

Answer 1

44 large houses and 16 small housed should be sold per month to obtain the best returns

Explanation:

Let x denotes large bird houses and y denotes small bird houses.

We are given that Both types of houses go through two technicians; first a shaper and then a painter

The large house requires two hours of the shaper's time and and then a painter.

The small house requires one hour of the shaper's time and two hours of the painter's time.

We are given that The shaper has 104 hours and the painter has 76 hours of time available this month.

So,
`2x+ y \leq 104`

`x+2y \leq76`

`x \geq 0`

`y \geq 0`

Plot the lines on the graph

Refer the attached graph

The boundary points of feasible region are :

(0,0),(0,38),(44,16),(52,0)

Profit on the large house is $6 and on the small house is $11.

So, P=6x+11y

At(0,0)

P=0

At(0,38)

P=6(0)+11(38)=418

At (44,16)

P=6(44)+11(16)=440

At(52,0)

P=6(52)+11(0)=312

So, The company gets the best returns at(44,16)

So, 44 large houses and 16 small housed should be sold per month to obtain the best returns