let y equal the number of pacific.
each Atlantic model requires 1000 feet of framing lumber, 3000 cubic feet of concrete, and 2000 for advertising.
each pacific model requires 2000 feet of framing lumber, 3000 cubic feet of concrete and $3000 for advertising.
contracts call for using at least 8000 feet of framing lumber, 18,000 cubic feet of concrete and $15000 worth of advertising.
if the total spent on each Atlantic model is $3000 and the total spent on each pacific model is $4000 how many of each model should be built to minimize costs?
your constraint functions are:
>= 8000 feet of framing lumber.
>= 18000 cubic feet of concrete
>= 15000 dollars of advertising.
each atlantic model requires 1000 feet of framing lumber.
each pacific model requires 2000 feet of framing lumber.
therefore 1000 * x + 2000 * y >= 8000
each atlantic model requires 3000 cubic feet of concrete.
each pacific model requires 3000 cubic feet of concrete.
therefore 3000 * x + 3000 * y >= 18000
each atlantic model requires 2000 dollars of advertising.
each pacific model requires 3000 dollars of advertising.
therefore 2000 * x + 3000 * y >= 15000
since the number of houses built can't be negative, then:
x >= 0
y >= 0
each atlantic model costs 3000 dollars to build and each pacific model costs 4000 dollars to build.
you want to minimize costs, so your objective function is:
cost = 3000 * x + 4000 * y
you will graph the constraint and then evaluate the corner points of the feasible region by the objective function to see which corner point gives you the minimum cost.
using the desmos.com calculator, you would graph the OPPOSITE of the inequalities and then find the region on the graph that is NOT shaded.
that's your feasible region.
you then find the corner points of the feasible region and analyze the cost function at those corner points.
your constraint functions are:
1000 * x + 2000 * y >= 8000
3000 * x + 3000 * y >= 18000
2000 * x + 3000 * y >= 15000
x >= 0
y >= 0
you will graph:
1000 * x + 2000 * y <= 8000
3000 * x + 3000 * y <= 18000
2000 * x + 3000 * y <= 15000
x <= 0
y <= 0
your graph will look like this:
the corner points of your feasible region are:
(0,6)
(3,3)
(6,1)
(8,0)
these coordinate points are in (x,y) format.
your cost function is 3000 * x + 4000 * y
at each of the corner points, your cost will be:
(0,6) = 0 * 3000 + 6 * 4000 = 24000
(3,3) = 3 * 3000 + 3 * 4000 = 21000 *****
(6,1) = 6 * 3000 + 1 * 4000 = 22000
(8,0) = 8 * 3000 + 0 * 4000 = 24000
***** your minimum cost is when you build 3 atlantic models and 3 pacific models.
all constraints have to be satisfied.
your constraints are analyzed when x = 3 and y = 3:
1000 * x + 2000 * y >= 8000 becomes 9000 >= 8000 = satisfied.
3000 * x + 3000 * y >= 18000 becomes 18000 >= 18000 = satisfied.
2000 * x + 3000 * y >= 15000 becomes 15000 >= 15000 = satisfied.
x >= 0 = satisfied
y >= 0 = satisfied
all constraints are satisfied, therefore the solution is good.
the solution is that the cost is minimized when x = 3 and y = 3 which means that 3 atlantic models and 3 pacific models are constructed.