Question 1071178
first of all, you want to convert everything to pounds or ounces.
i chose to convert all to pounds.


1 pound is equal to 16 ounces, therefore, divide the number of ounces by 16 and you get the number of pounds.


you are given that the saltier blend contains 4 ounces of twilight delight chocolate and 12 ounces of dark chocolate with salt.


translate to pounds and you get that the saltier blend contains .25 pounds of twilight delight chocolate and .75 pounds of dark chocolate with salt.


you are given that the other blend contains 8 ounces of twilight delight chocolate and 8 ounces of dark chocolate with salt.


translate to pounds and you get that the other blend contains .5 pounds of twilight delight chocolate and .5 pounds of dark chocolate with salt.


you are also given that the number of available pounds of the dark chocolate with salt is equal to 120 and that the number of available pounds of twilight delight chocolate is equal to 100.


let x = the number of pounds of the saltier blend and let y = the number of pounds of the other blend.


your blends are sold in 1 pound packages.


your profit is .3 per package of the saltier blend and .4 per package of the other blend.


this translates to .3 per pound of the saltier blend and .4 per pound of the other blend.


your profit will therefore = .3 * number of pounds of the saltier blend + .4 * number of pounds of the other blend.


algebraically, this becomes profit = .3x + .4y.


that becomes your objective function because that's what you want to maximize.


your constraints will be:


x >= 0
y >= 0


this is because the number of pounds of either product can't be negative.


.25x + .5y <= 100


this is because the number of pounds of twilight delight chocolate must be less than or equal to 100 pounds and this is composed of .25 pounds of chocolate delight for each pound of the saltier blend and .5 pounds of chocolate delight for each pound of the other blend.


.75x + .5y <= 120


this is because the number of pounds of dark chocolate with salt must be less than or equal to 120 pounds and this is composed of .75 pounds of dark chocolate with salt for each pound of the saltier blend and .5 pounds of dark chocolate with salt for each pound of the other blend.


your objective function is:


.3x + .4y = profit.


your constraint functions are:


x >= 0
y >= 0
.25x + .5y <= 100
.75x + .5y <= 120


using the www.desmos.com/calculator, you would graph the opposite inequalities as shown below:


x <= 0
y <= 0
.25x + .5y >= 100
.75x + .5y >= 120


the area of the graph that is NOT shaded is your region of feasibility.


you would then evaluate the objective function at each of these corner points.


your maximum profit will be at one of those corner points.


your graph will look like this:


<img src = "http://theo.x10hosting.com/2017/030302.jpg" alt="$$$" </>


at the point (0,200), your profit is .4 * 200 = 80.
at the point (40,180), your profit is .3 *40 + .4 * 180 = 84.
at the point (160,0), your profits is .3 * 160 = 48.


your maximum profit is at the point (40,180).


this means you sell 40 pounds of the saltier blend and 180 pounds of the other blend.


your constraints need to be met at the point (40,180).


x and y >= 0 are met because x = 40 and y = 180.


.25x + .5y <= 100 is met because .25*40 + .5*180 = 100.


.75x + .5y <= 120 is met because .75*40 + .5*180 = 120.


if you were to do this manually, you would graph the equalities and then find the area on the graph that satisfied the inequalities.


you would graph:


x = 0
y = 0
.25x + .5y = 100
.75x + .5y = 120


you would then find the area on the graph that satisfies the inequality.


the inequalities are:


x >= 0
y >= 0
.25x + .5y <= 100
.75x + .5y <= 120


in that particular case, your graph would look like this:


<img src = "http://theo.x10hosting.com/2017/030303.jpg" alt="$$$" </>


in this case, the area that IS shaded is your region of feasibility.


note also that most graphing software require you to solve for y before graphing.


for example:


in the equation of .25x + .5y = 100, you would have to solve for y first.


your equation would become y = (100 - .25x) / .5.


that's the equation that you would graph.


it's the same equation so you'll get the same graph, but desmos.com allows you to graph is as .25x + .5y = 100 while other graphing software require you to graph it as y = (100 - .25x) / .5.


in fact, in most graphing software, the y is assumed, so you would graph (100 - .25x) / .5.