Question 1206298
x = number of small boxes.
y = number of large boxes.


constraint inequalities are:
y <= 20
x <= 30
.8x + 1.4y <= 42
.5x + .5y <= 24
x >= 0]
y >= 0
objective function:
profit = 42x + 14y


using the decmos.com calculator, you would graph the opposite of the inequalities.
the feasible region is the area on the graph that is not shaded.
you would evaluate the objective function at each corner point to find the maximum profit.
since partial boxes are not allowed, you would round to the next lowest integer.


the profit at each corner point is shown below.


(0,20) = 840
(17.5,20) = (17,20) = 994
(30,12.857) = (30,12) = 1428
(30,0) = 1260


the maximum profit is at (20,12) where profit is equal to 42 * 30 + 12 * 14 = 1428.

all constraints are satisfied, such as:
wood constraint at .8 * 30 + 1.4 * 12 = 40.8 <= 42 is satisfied.
labor constraint at .5 * 30 + .5 * 12 = 21 <= 24 is satisfied.
0 <= x <= 30 is satisfied.
0 <= y <= 20 is satisfied.


here's what the graph looks like:


<img src = "http://theo.x10hosting.com/2024/022801.jpg">