your objective function is profit = 2x + 2.5y
your constraint functions are:
20x + 10y <= 40,000
15x + 30y <= 60,000
10x + 12y <= 27,000
x >= 0
y >= 0
using the desmos.com calculator, you would graph the opposite of these constraint functions.
you would graph:
20x + 10y >= 40,000
15x + 30y >= 60,000
10x + 12y >= 27,000
x <= 0
y <= 0
the area of the graph that is not shaded is your region of feasibility.
your maximum profit is at the corner points of this graph.
you evaluate your objective function at each of these corner points.
you will find that the maximum profit is at the (750,1625) corner point, where your profit will be 2 * 750 + 2.5 * 1625 = 5,562.5 dollars.
you will have utilized ...
20 * 750 + 10 * 1625 = 31,250 peaches,
15 * 750 + 20 * 1625 = 60,000 apples,
10 * 750 + 12 * 1625 = 27,000 pears.
it appears that all of your apples and pears will have been utilized.
you will have 40,000 - 31,250 = 8,750 apples left over.
it helps to set up a table to assist you in figuring out what to do.
your table might look something like this:
package A package B additional information
x y number of each type package
peaches 20x + 10y <= 40,000
apples 15x + 30y <= 60,000
pears 10x + 12y <= 27,000
profit 2x + 2.5y maximize
this might help you visualize how to set up the problem.
once you get the hang of it, the table is not always necessary, but i still use some form of it just to set my mind straight.
it does help.
not all calculators allow you to solve the problem this way.
most require you to solve for y first.
most don't do inequalities.
you can use these calculators to graph what you can and then manually shade the area of the graph that satisfies the inequalities plus graph any functions that the calculator can't handle.
in that case, you would do the following:
solve for y in each of the constraint functions that you can.
graph the equality portion of each of these constraint functions.
shade the area of the graph that satisfies the inequalities.
your objective function is profit = x * 2.00 + y * 2.50
your constraint functions are:
20x + 10y <= 40,000
15x + 30y <= 60,000
10x + 12y <= 27,000
x >= 0
y >= 0
you would graph:
y = (40,000-20x)/10
y = (60,000-15x)/30
y = (27,000-10x)/12
y = 0
x = 0 (might have to be done manually)
you would then manually shade the area of the graph that satisfies the inequalities.
your graph would look like something like this:
